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❘✐s❦ ❆♥❛❧②t✐❝s ❆✉t✉♠♥ ✷✵✶✻

❈♦❧✐♥ ❘♦✇❛t ❝✳r♦✇❛t❅❜❤❛♠✳❛❝✳✉❦ ❖❝t♦❜❡r ✹✱ ✷✵✶✻ ✭♣r❡❧✐♠✐♥❛r② ✉♥t✐❧ ❡♥❞ ♦❢ t❡r♠✮

✶ ✴ ✷✷✹

❈♦♥t❡♥ts

■♥tr♦❞✉❝t✐♦♥

✶

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s

✷

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s

✸

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t

✹

❊st✐♠❛t✐♥❣ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts

✺

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s

✻

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s

✼

❊st✐♠❛t✐♥❣ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦

✽

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦

✾

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦

✶✵

❘❡❣✉❧❛t♦r② ❢r❛♠❡✇♦r❦ ♦❢ r✐s❦ ♠❛♥❛❣❡♠❡♥t ✷ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥

■♥tr♦❞✉❝t✐♦♥

✜♥❛♥❝✐❛❧ ❛ss❡ts ❛r❡ ❤❡❧❞ ♦✉t ♦❢ ❛♥ ✐♥t❡r❡st ✐♥ t❤❡✐r ❢✉t✉r❡ ♣❛②♦✛s ❢✉t✉r❡ ♣❛②♦✛s ❛r❡ r✐s❦② ❤❡♥❝❡✱ ❛ss❡t ♠❛♥❛❣❡♠❡♥t ♥❡❝❡ss❛r✐❧② ✐♥✈♦❧✈❡s r✐s❦ ♠❛♥❛❣❡♠❡♥t t❤❡ ❥♦❜ ♠❛r❦❡t ❤✐❣❤❧② ✈❛❧✉❡s r✐s❦ ♠❛♥❛❣❡♠❡♥t ❛❜✐❧✐t✐❡s ✵✶✳✶✵✳✶✺ ✶✾✳✵✾✳✶✻ r✐s❦ ❥♦❜s ✻✻✺ ✺✶✷ t♦t❛❧ ❥♦❜s ✶✱✷✺✼ ✾✷✹

❚❛❜❧❡✿ ❵◗✉❛♥t✐t❛t✐✈❡ ✜♥❛♥❝❡✬ s❡❛r❝❤❡s ♦♥ ❤tt♣✿✴✴✇✇✇✳✐♥❞❡❡❞✳❝♦♠

❘✐s❦ ♠❛♥❛❣❡rs ❛r❡ ✐♥ ❣r❡❛t ❞❡♠❛♥❞ ❛s ❛ r❡s✉❧t ♦❢ t❤❡ tr♦✉❜❧❡s t❤❡ ❜❛♥❦s ❤❛✈❡ ❢♦✉♥❞ t❤❡♠s❡❧✈❡s ✐♥ ✭❘✐❝❤❛r❞ ▲✐♣st❡✐♥✱ ❇♦②❞❡♥ ●❧♦❜❛❧ ❊①❡❝✉t✐✈❡ ❙❡❛r❝❤✮ ❇❛♥❦s ❛r❡ ❢❛✐❧✐♥❣ t♦ ✐♠♣❧❡♠❡♥t ❜♦♥✉s ♣❧❛♥s t❤❛t r❡✐♥ ✐♥ t❤❡ t②♣❡s ♦❢ r✐s❦s ❜❧❛♠❡❞ ❢♦r ❝♦♥tr✐❜✉t✐♥❣ t♦ t❤❡ ✜♥❛♥❝✐❛❧ ❝r✐s✐s✱ ❬t❤❡ ❇❛s❡❧ ❈♦♠♠✐tt❡❡ ♦♥ ❇❛♥❦✐♥❣ ❙✉♣❡r✈✐s✐♦♥❪ s❛✐❞✳ ❇❛♥❦ ❇♦♥✉s P❧❛♥s ❋❛✐❧ t♦ ❈✉r❜ ❋✐♥❛♥❝✐❛❧ ❘✐s❦s✱ ❘❡❣✉❧❛t♦rs ❙❛② ✭❇❧♦♦♠❜❡r❣✱ ✶✺ ❖❝t ✷✵✶✵✮

✸ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❈❛✈❡❛t ♠❡r❝❛t♦r

❖✉r ✭❛❤❡♠✦✮ ♣r♦♠✐s❡ t♦ ②♦✉

❆♥② ❣r❛❞✉❛t❡ ♦❢ ●✺✸ ❘✐s❦ ❆♥❛❧②t✐❝s s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ ❜❡❛t ❛♥ ✐♥❞❡① ❢✉♥❞ ♦✈❡r t❤❡ ❝♦✉rs❡ ♦❢ ❛ ♠❛r❦❡t ❝②❝❧❡✳ ♦✉t♣❡r❢♦r♠❡❞ ▼❙❈■❄ ♠❡t ❱❛❘❄ ②❡s ✻ ✶ ♥♦ ✻ ✶ ❄ ✷ ✶✷

❚❛❜❧❡✿ ✷✵✶✵✲✶✶ st✉❞❡♥ts✬ ♣❡r❢♦r♠❛♥❝❡ ♦♥ ●✺✸ ♣♦rt❢♦❧✐♦ ♦♣t✐♠✐s❛t✐♦♥ ❛ss✐❣♥♠❡♥t

❚❤✐s ❛rt✐❝❧❡ ✐s ❝❛✉t✐♦♥❛r②✿ r❡❛❞ ✐t✱ ❛♥❞ t❛❦❡ ✐t s❡r✐♦✉s❧②✳ ❙❡❡ ❛❧s♦ ❇♦♦❦st❛❜❡r ✭✷✵✵✼✮ ❢♦r ❛❝❝♦✉♥ts ♦❢ s♠❛rt✱ t❛❧❡♥t❡❞✱ ❤❛r❞✲✇♦r❦✐♥❣ ♣❡♦♣❧❡ ❧♦s✐♥❣ ✈❛st q✉❛♥t✐t✐❡s ♦❢ ♠♦♥❡②✳ ❆r❡ t❤❡ t❡❝❤♥✐q✉❡s t❛✉❣❤t ✐♥ ●✺✸ t❤❡ ♦♣♣♦s✐t❡ ♦❢ t❤❡ ❵✈❛❧✉❡✬ t❡❝❤♥✐q✉❡s ✉s❡❞ ❜② ✐♥✈❡st♦rs ❧✐❦❡ ❇❡♥ ●r❛❤❛♠ ❛♥❞ ❲❛rr❡♥ ❇✉✛❡tt ✭❙❝❤r♦❡❞❡r✱ ✷✵✵✾✮✱ ♦r ❛♥ ❛✉❣♠❡♥t❛t✐♦♥ ♦❢ t❤❡♠❄

✺ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❈❛✈❡❛t ♠❡r❝❛t♦r

❆ ❧✐tt❧❡ ▲❡❛r♥✐♥❣ ✐s ❛ ❞❛♥❣✬r♦✉s ❚❤✐♥❣

■❢ ✏❛❝t✐✈❡✑ ❛♥❞ ✏♣❛ss✐✈❡✑ ♠❛♥❛❣❡♠❡♥t st②❧❡s ❛r❡ ❞❡✜♥❡❞ ✐♥ s❡♥s✐❜❧❡ ✇❛②s✱ ✐t ♠✉st ❜❡ t❤❡ ❝❛s❡ t❤❛t

✶

❜❡❢♦r❡ ❝♦sts✱ t❤❡ r❡t✉r♥ ♦♥ t❤❡ ❛✈❡r❛❣❡ ❛❝t✐✈❡❧② ♠❛♥❛❣❡❞ ❞♦❧❧❛r ✇✐❧❧ ❡q✉❛❧ t❤❡ r❡t✉r♥ ♦♥ t❤❡ ❛✈❡r❛❣❡ ♣❛ss✐✈❡❧② ♠❛♥❛❣❡❞ ❞♦❧❧❛r ❛♥❞

✷

❛❢t❡r ❝♦sts✱ t❤❡ r❡t✉r♥ ♦♥ t❤❡ ❛✈❡r❛❣❡ ❛❝t✐✈❡❧② ♠❛♥❛❣❡❞ ❞♦❧❧❛r ✇✐❧❧ ❜❡ ❧❡ss t❤❛♥ t❤❡ r❡t✉r♥ ♦♥ t❤❡ ❛✈❡r❛❣❡ ♣❛ss✐✈❡❧② ♠❛♥❛❣❡❞ ❞♦❧❧❛r ❚❤❡s❡ ❛ss❡rt✐♦♥s ✇✐❧❧ ❤♦❧❞ ❢♦r ❛♥② t✐♠❡ ♣❡r✐♦❞✳ ▼♦r❡♦✈❡r✱ t❤❡② ❞❡♣❡♥❞ ♦♥❧② ♦♥ t❤❡ ❧❛✇s ♦❢ ❛❞❞✐t✐♦♥✱ s✉❜tr❛❝t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥✳ ◆♦t❤✐♥❣ ❡❧s❡ ✐s r❡q✉✐r❡❞✳ ✭❙❤❛r♣❡✱ ✶✾✾✶✮

♠❛r❦❡t r❡t✉r♥✿ ✏✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♦❢ t❤❡ r❡t✉r♥s ♦♥ t❤❡ ❛❝t✐✈❡ ❛♥❞ ♣❛ss✐✈❡ s❡❣♠❡♥ts ♦❢ t❤❡ ♠❛r❦❡t✑ ❛✈❡r❛❣❡ ♣❛ss✐✈❡ r❡t✉r♥✿ s❛♠❡ ❛s t❤❡ ♠❛r❦❡t r❡t✉r♥ ❤❡♥❝❡ ❛✈❡r❛❣❡ ❛❝t✐✈❡ r❡t✉r♥ ✐s t♦♦ ❛❝t✐✈❡ ♠❛♥❛❣❡♠❡♥t ❢❛❝❡s ❤✐❣❤❡r ❝♦sts r❡❝❡♥t r❡♠✐♥❞❡r✿ ❆❝t✐✈❡ ❢✉♥❞s ✉♥❞❡r♣❡r❢♦r♠ t❤❡ ❵✐♥❡rt✐❛ ✐♥❞❡①✬✱ ♥❡✈❡r ♠✐♥❞ t❤❡ ♠❛r❦❡t ✭✷✼✴✵✾✴✶✺✱ ❋❚✮

✻ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❆ st❛♥❞❛r❞ r✐s❦ t❛①♦♥♦♠②

▼❛r❦❡t r✐s❦

❚❤❡ ❜❡st ❦♥♦✇♥ t②♣❡ ♦❢ r✐s❦ ✐s ♣r♦❜❛❜❧② ♠❛r❦❡t r✐s❦✿ t❤❡ r✐s❦ ♦❢ ❛ ❝❤❛♥❣❡ ✐♥ t❤❡ ✈❛❧✉❡ ♦❢ ❛ ✜♥❛♥❝✐❛❧ ♣♦s✐t✐♦♥ ❞✉❡ t♦ ❝❤❛♥❣❡s ✐♥ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ❝♦♠♣♦♥❡♥ts ♦♥ ✇❤✐❝❤ t❤❛t ♣♦rt❢♦❧✐♦ ❞❡♣❡♥❞s✱ s✉❝❤ ❛s st♦❝❦ ❛♥❞ ❜♦♥❞ ♣r✐❝❡s✱ ❡①❝❤❛♥❣❡ r❛t❡s✱ ❝♦♠♠♦❞✐t② ♣r✐❝❡s✱ ❡t❝✳ ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✺✮ ❘❡st✐ ❛♥❞ ❙✐r♦♥✐ ✭✷✵✵✼✱ P❛rt ■■✮

✽ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❆ st❛♥❞❛r❞ r✐s❦ t❛①♦♥♦♠②

❈r❡❞✐t r✐s❦

❚❤❡ ♥❡①t ✐♠♣♦rt❛♥t ❝❛t❡❣♦r② ✐s ❝r❡❞✐t r✐s❦✿ t❤❡ r✐s❦ ♦❢ ♥♦t r❡❝❡✐✈✐♥❣ ♣r♦♠✐s❡❞ r❡♣❛②♠❡♥ts ♦♥ ♦✉tst❛♥❞✐♥❣ ✐♥✈❡st♠❡♥ts s✉❝❤ ❛s ❧♦❛♥s ❛♥❞ ❜♦♥❞s✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ ✏❞❡❢❛✉❧t✑ ♦❢ t❤❡ ❜♦rr♦✇❡r✳ ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✺✮ r❡✢❡❝ts✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ✏✉♥❡①♣❡❝t❡❞ ❝❤❛♥❣❡s ✐♥ t❤❡ ❝r❡❞✐t✇♦rt❤✐♥❡ss ♦❢ ❬❛ ❜❛♥❦✬s❪ ❝♦✉♥t❡r♣❛rt✐❡s✑ ✭❘❡st✐ ❛♥❞ ❙✐r♦♥✐✱ ✷✵✵✼✱ P❛rt ■■■✮❀ t❤❡② ❢❡❡❧ ✐t t♦ ❜❡ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝❧❛ss s❡❡ ❛❧s♦ ●♦r❞② ✭✷✵✵✵✮ ♦r ❇✐❡❧❡❝❦✐ ❛♥❞ ❘✉t❦♦✇s❦✐ ✭✷✵✵✷✮ ❋r❡❞❞✐❡ ▼❛❝ ❝♦✉❧❞ ✐♥❝✉r ✏❜✐❧❧✐♦♥s ♦❢ ❞♦❧❧❛rs ♦❢ ❧♦ss❡s✑ ❢♦r ❯❙ t❛①♣❛②❡rs ❜② ❢♦❝✉ss✐♥❣ ♦♥ ♠♦rt❣❛❣❡s ❢❛✐❧✐♥❣ ✐♥ t❤❡ ✜rst t✇♦ ②❡❛rs❀ ✇❤✐❧❡✱ ❤✐st♦r✐❝❛❧❧②✱ t❤✐s ✐s ✇❤❡♥ ♠♦st ❤❛✈❡ ❢❛✐❧❡❞✱ ♠♦r❡ r❡❝❡♥t ❵t❡❛s❡r✬ ♠♦rt❣❛❣❡s ❤❛✈❡ ❧♦✇ ✐♥t❡r❡st r❛t❡s ❢♦r t❤r❡❡ t♦ ✜✈❡ ②❡❛rs✱ ✇❤✐❝❤ t❤❡♥ r✐s❡ s❤❛r♣❧② ✭❋r❡❞❞✐❡ ▼❛❝ ▲♦❛♥ ❉❡❛❧ ❉❡❢❡❝t✐✈❡✱ ❘❡♣♦rt ❙❛②s✱ ◆❨❚✱ ✷✼ ❙❡♣t ✷✵✶✶✮

✾ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❆ st❛♥❞❛r❞ r✐s❦ t❛①♦♥♦♠②

❖♣❡r❛t✐♦♥❛❧ r✐s❦

❆ ❢✉rt❤❡r r✐s❦ ❝❛t❡❣♦r② ✐s ♦♣❡r❛t✐♦♥❛❧ r✐s❦✿ t❤❡ r✐s❦ ♦❢ ❧♦ss❡s r❡s✉❧t✐♥❣ ❢r♦♠ ✐♥❛❞❡q✉❛t❡ ♦r ❢❛✐❧❡❞ ✐♥t❡r♥❛❧ ♣r♦❝❡ss❡s✱ ♣❡♦♣❧❡ ❛♥❞ s②st❡♠s✱ ♦r ❢r♦♠ ❡①t❡r♥❛❧ ❡✈❡♥ts✳ ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✺✮ ❘❡st✐ ❛♥❞ ❙✐r♦♥✐ ✭✷✵✵✼✱ P❛rt ■❱✮ ❚❤❡ ❧♦ss r❡s✉❧t❡❞ ❢r♦♠ ✉♥❛✉t❤♦r✐s❡❞ s♣❡❝✉❧❛t✐✈❡ tr❛❞✐♥❣ ✐♥ ✈❛r✐♦✉s ❙✫P ✺✵✵✱ ❉❛① ❛♥❞ ❊✉r♦st♦①① ✐♥❞❡① ❢✉t✉r❡s ♦✈❡r t❤❡ ♣❛st t❤r❡❡ ♠♦♥t❤s✳ ❚❤❡ ♣♦s✐t✐♦♥s ❤❛❞ ❜❡❡♥ ♦✛s❡t ✐♥ ♦✉r s②st❡♠s ✇✐t❤ ✜❝t✐t✐♦✉s✱ ❢♦r✇❛r❞✲s❡tt❧✐♥❣✱ ❝❛s❤ ❊❚❋ ♣♦s✐t✐♦♥s✱ ❛❧❧❡❣❡❞❧② ❡①❡❝✉t❡❞ ❜② t❤❡ tr❛❞❡r✳ ❚❤❡s❡ ✜❝t✐t✐♦✉s tr❛❞❡s ❝♦♥❝❡❛❧❡❞ t❤❡ ❢❛❝t t❤❛t t❤❡ ✐♥❞❡① ❢✉t✉r❡s tr❛❞❡s ✈✐♦❧❛t❡❞ ❯❇❙✬s r✐s❦ ❧✐♠✐ts✳ ✭❯❇❙✱ ✶✽ ❙❡♣t❡♠❜❡r ✷✵✶✶✮

✶✵ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❆ st❛♥❞❛r❞ r✐s❦ t❛①♦♥♦♠②

▼♦❞❡❧ r✐s❦

▼♦❞❡❧ r✐s❦ ♠❛♥❛❣❡♠❡♥t ❤❛s ❜❡❝♦♠❡ ❛ ❜♦❛r❞✲❧❡✈❡❧ ♣r♦❝❡ss✳ ◆♦✇ t❤❡ ❝❤✐❡❢ r✐s❦ ♦✣❝❡r ❤❛s t♦ ❣♦ t♦ t❤❡ ❜♦❛r❞ ❛♥❞ ♥♦t ♦♥❧② t❛❧❦ ❛❜♦✉t ♠❛r❦❡t r✐s❦✱ ❝r❡❞✐t r✐s❦ ❛♥❞ ♦♣❡r❛t✐♦♥❛❧ r✐s❦✱ ❤❡ ❛❧s♦ ❤❛s t♦ t❛❧❦ ❛❜♦✉t ♠♦❞❡❧ r✐s❦✳ ■t ✐s ❛ ❤✉❣❡ ♦r❣❛♥✐s❛t✐♦♥❛❧ ❝❤❛♥❣❡✳ ✭◆❡✇ ❨♦r❦✲❜❛s❡❞ ♠♦❞❡❧ r✐s❦ ♠❛♥❛❣❡r❀ ❙❤❡r✐❢ ✭✷✵✶✻✮✮ ❛r✐s❡s ❢r♦♠ ❛ ♠✐ss♣❡❝✐✜❡❞ ♠♦❞❡❧ ❡✳❣✳ ✉s✐♥❣ ❇❧❛❝❦✲❙❝❤♦❧❡s ✇❤❡♥ ♠♦❞❡❧ ❛ss✉♠♣t✐♦♥s ❞♦♥✬t ❤♦❧❞ ✭❡✳❣✳ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ r❡t✉r♥s✮ ✏❛❧✇❛②s ♣r❡s❡♥t t♦ s♦♠❡ ❞❡❣r❡❡✑ ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✺✮ q✳✈✳ ❘❡❜♦♥❛t♦ ✭✷✵✵✼✮✱ ❙✉♣❡r✈✐s♦r② ●✉✐❞❛♥❝❡ ♦♥ ♠♦❞❡❧ r✐s❦ ♠❛♥❛❣❡♠❡♥t ✭✷✵✶✶✮✱ ▼♦r✐♥✐ ✭✷✵✶✶✮

✶✶ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❆ st❛♥❞❛r❞ r✐s❦ t❛①♦♥♦♠②

▲✐q✉✐❞✐t② r✐s❦

❲❤❡♥ ✇❡ t❛❧❦ ❛❜♦✉t ❧✐q✉✐❞✐t② r✐s❦ ✇❡ ❛r❡ ❣❡♥❡r❛❧❧② r❡❢❡rr✐♥❣ t♦ ♣r✐❝❡ ♦r ♠❛r❦❡t ❧✐q✉✐❞✐t② r✐s❦✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❜r♦❛❞❧② ❞❡✜♥❡❞ ❛s t❤❡ r✐s❦ st❡♠♠✐♥❣ ❢r♦♠ t❤❡ ❧❛❝❦ ♦❢ ♠❛r❦❡t❛❜✐❧✐t② ♦❢ ❛♥ ✐♥✈❡st♠❡♥t t❤❛t ❝❛♥♥♦t ❜❡ ❜♦✉❣❤t ♦r s♦❧❞ q✉✐❝❦❧② ❡♥♦✉❣❤ t♦ ♣r❡✈❡♥t ♦r ♠✐♥✐♠✐③❡ ❛ ❧♦ss✳ ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✺✮ ■♥ ❜❛♥❦✐♥❣✱ t❤❡r❡ ✐s ❛❧s♦ t❤❡ ❝♦♥❝❡♣t ♦❢ ❢✉♥❞✐♥❣ ❧✐q✉✐❞✐t② r✐s❦✱ ✇❤✐❝❤ r❡❢❡rs t♦ t❤❡ ❡❛s❡r ✇✐t❤ ✇❤✐❝❤ ✐♥st✐t✉t✐♦♥s ❝❛♥ r❛✐s❡ ❢✉♥❞✐♥❣ t♦ ♠❛❦❡ ♣❛②♠❡♥ts ❛♥❞ ♠❡❡t ✇✐t❤❞r❛✇❛❧s ❛s t❤❡② ❛r✐s❡✳ ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✺✮

✶✷ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼❡✉❝❝✐ ♠❛♥tr❛

❚❤❡ ▼❡✉❝❝✐ ♠❛♥tr❛

✶ ❢♦r ❡❛❝❤ s❡❝✉r✐t②✱ ✐❞❡♥t✐❢② t❤❡ ✐✐❞ st♦❝❤❛st✐❝ t❡r♠s ✭➓✸✳✶✮ ✷ ❡st✐♠❛t❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts ✭➓✹✮ ✸ ♣r♦❥❡❝t t❤❡ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ ✭➓✸✳✷✮ ✹ ❞✐♠❡♥s✐♦♥ r❡❞✉❝❡ t♦ ♠❛❦❡ t❤❡ ♣r♦❜❧❡♠ ♠♦r❡ tr❛❝t❛❜❧❡ ✭➓✸✳✹✮ ✺ ❡✈❛❧✉❛t❡ t❤❡ ♣♦rt❢♦❧✐♦ ♣❡r❢♦r♠❛♥❝❡ ❛t t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ ✭➓✺✮

✇❤❛t ✐s ②♦✉r ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥❄

✻ ♣✐❝❦ t❤❡ ♣♦rt❢♦❧✐♦ t❤❛t ♦♣t✐♠✐s❡s ②♦✉r ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✭➓✻✮ ✼ ❛❝❝♦✉♥t ❢♦r ❡st✐♠❛t✐♦♥ r✐s❦ ✶

r❡♣❧❛❝❡ ♣♦✐♥t ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s ✇✐t❤ ❇❛②❡s✐❛♥ ❞✐str✐❜✉t✐♦♥s ✭➓✼✮

✷

r❡✲❡✈❛❧✉❛t❡ t❤❡ ♣♦rt❢♦❧✐♦ ❞✐str✐❜✉t✐♦♥s ✐♥ t❤✐s ❧✐❣❤t ✭➓✽✮

✸

r♦❜✉st❧② r❡✲♦♣t✐♠✐s❡ ✭➓✾✮

❖❜s❡r✈❛t✐♦♥ s❤♦✇s t❤❛t s♦♠❡ st❛t✐st✐❝❛❧ ❢r❡q✉❡♥❝✐❡s ❛r❡✱ ✇✐t❤✐♥ ♥❛rr♦✇❡r ♦r ✇✐❞❡r ❧✐♠✐ts✱ st❛❜❧❡✳ ❇✉t st❛❜❧❡ ❢r❡q✉❡♥❝✐❡s ❛r❡ ♥♦t ✈❡r② ❝♦♠♠♦♥✱ ❛♥❞ ❝❛♥♥♦t ❜❡ ❛ss✉♠❡❞ ❧✐❣❤t❧②✳ ❑❡②♥❡s ✭✶✾✷✶✱ ♣✳✸✽✶✮

✶✹ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼❡✉❝❝✐ ♠❛♥tr❛

◆♦t❛t✐♦♥❛❧ ❝♦♥✈❡♥t✐♦♥s

τ✱ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ T✱ t✐♠❡ ❛t ✇❤✐❝❤ t❤❡ ❛❧❧♦❝❛t✐♦♥ ❞❡❝✐s✐♦♥ ✐s ♠❛❞❡

t❤✉s✱ T + τ ✐s ✇❤❡♥ t❤❡ ✐♥✈❡st♠❡♥ts ❛r❡ t♦ ❜❡ ❡✈❛❧✉❛t❡❞

Pt✱ t❤❡ ✈❡❝t♦r ♦❢ ♣r✐❝❡s ❛t t✐♠❡ t Xt✱ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ t❤❛t ✇✐❧❧ r❡❛❧✐s❡ ❛t t✐♠❡ t xt✱ ❛ r❡❛❧✐s❛t✐♦♥ ♦❢ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ iT ≡ {x✶, . . . , xT}✱ ❛ ❞❛t❛s❡t ♦❢ ♦❜s❡r✈❡❞ r❡❛❧✐s❛t✐♦♥s

✶✺ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥

❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s

❛ ♥✉♠❜❡r ✇❤♦s❡ r❡❛❧✐s❛t✐♦♥ ✐s✱ ❛s ②❡t✱ ✉♥❦♥♦✇♥ ✐ts ❞✐str✐❜✉t✐♦♥ ♠❛② ❜❡ ❦♥♦✇♥

❛ s♣❛❝❡ ♦❢ ❡✈❡♥ts✱ E ❛ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✱ P

❛ ❢✉♥❝t✐♦♥ ❢r♦♠ t❤❡ s♣❛❝❡ ♦❢ ❡✈❡♥ts t♦ t❤❡ r❡❛❧ ❧✐♥❡ t❤✉s✱ x = X (e) ❢♦r s♦♠❡ ❡✈❡♥t e ✐♥ E t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥ ❡✈❡♥t ❣✐✈✐♥❣ r✐s❡ t♦ ❛ r❡❛❧✐s❡❞ x ∈ [x, ¯ x]✿ P {X ∈ [x, ¯ x]} ≡ P {e ∈ E s✳t✳ X (e) ∈ [x, ¯ x]} . r❡❛❞s✿ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ X t❛❦❡s ♦♥ ❛ ✈❛❧✉❡ ✐♥ [x, ¯ x] ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ s❡t ♦❢ ❡✈❡♥ts ②✐❡❧❞✐♥❣ ❛ r❡❛❧✐s❡❞ ✈❛❧✉❡ ♦❢ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✳ ✳ ✳ ❣♦✐♥❣ ❢♦r✇❛r❞✱ t②♣✐❝❛❧❧② s✉♣♣r❡ss ❞❡♣❡♥❞❡♥❝❡ ♦♥ e✱ r❡❢❡r ❥✉st t♦ X

♥❛ï✈❡ st❛t❡♠❡♥t ✶✼ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥

Pr♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭P❉❋✮✱ fX

✶ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ r✈ X

t❛❦❡s ♦♥ ❛ ✈❛❧✉❡ ✇✐t❤✐♥ ❛ ❣✐✈❡♥ ✐♥t❡r✈❛❧ P {X ∈ [x, ¯ x]} ≡ ¯

x x

fX (x) dx

✷ ✇❤② ❞♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧s♦ ❤♦❧❞❄

fX (x) ≥ ✵ ∞

−∞

fX (x) dx = ✶

❊①❛♠♣❧❡

fX (x) = ✶ √πe−x✷

−✸ −✷ −✶ ✵ ✶ ✷ ✸ ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ① ②

✶✽ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥

❈✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✭❈❉❋✮✱ FX

✶ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ r✈ X

t❛❦❡s ♦♥ ❛ ✈❛❧✉❡ ❧❡ss t❤❛♥ x FX (x) ≡ P {X ≤ x} = x

−∞

fX (u) du

✷ ✇❤② ❞♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧s♦ ❤♦❧❞❄

FX (−∞) = ✵ FX (∞) = ✶ FX♥♦♥✲❞❡❝r❡❛s✐♥❣

❊①❛♠♣❧❡

FX (x) = ✶ ✷ [✶ + ❡r❢ (x)] ✇❤❡r❡ ❡r❢ ✐s t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ ❡r❢ (x) ≡ ✷ √π x

✵

e−t✷dt

✶✾ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥

❈❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ✭❝❢✮✱ φX

✶

φX (ω) ≡ E

• eiωX

, ω ∈ R

✷ ✐ts ♣r♦♣❡rt✐❡s ❛r❡ ❧❡ss ✐♥t✉✐t✐✈❡

✭▼❡✉❝❝✐✱ ✷✵✵✺✱ q✳✈✳ ♣♣✳✻✲✼✮

✸ ♣❛rt✐❝✉❧❛r❧② ✉s❡❢✉❧ ❢♦r ❤❛♥❞❧✐♥❣

✭✇❡✐❣❤t❡❞✮ s✉♠s ♦❢ ✐♥❞❡♣❡♥❞❡♥t r✈s

❊①❛♠♣❧❡

φX (ω) = e− ✶

✷ω✷

−✸ −✷ −✶ ✵ ✶ ✷ ✸ ✵✳✵ ✵✳✷ ✵✳✹ ✵✳✻ ✵✳✽ ✶✳✵ ♦♠❡❣❛ ♣❤✐

✷✵ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥

◗✉❛♥t✐❧❡✱ QX

✶ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❈❉❋

QX (p) ≡ F −✶

X

(p)

✷ t❤❡ ♥✉♠❜❡r x s✉❝❤ t❤❛t t❤❡

♣r♦❜❛❜✐❧✐t② t❤❛t X ❜❡ ❧❡ss t❤❛♥ x ✐s p✿ P {X ≤ QX (p)} = p

❊①❛♠♣❧❡

QX (p) = ❡r❢−✶ (✷p − ✶)

❊①❛♠♣❧❡

❋♦r t❤❡ ♠❡❞✐❛♥✱ p = ✶

✷

❱❛❘ ✷✶ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥

❚❤❡ q✉❛♥t✐❧❡ ❛♥❞ t❤❡ ❈❉❋

x p = FX (x) p p′ x′ = QX (p′) x′ ✐♥✈❡rt✐❜✐❧✐t② r❡q✉✐r❡s fX > ✵ ♦t❤❡r✇✐s❡✱ ❝❛♥ r❡❣✉❧❛r✐s❡ fX ✇✐t❤ fX;ε ▼❡✉❝❝✐ ✭✷✵✵✺✱ ❆♣♣✳ ❇✳✹✮

✷✷ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥

▼♦✈✐♥❣ ❜❡t✇❡❡♥ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ r✈ X

QX ✐♥✈❡rs❡s FX D fX I I ◦ F−✶ F ◦ D φX F−✶ F I ✐s t❤❡ ✐♥t❡❣r❛t✐♦♥ ♦♣❡r❛t♦r D ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦♣❡r❛t♦r F ✐s t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭❋❚✮ F−✶ ✐s t❤❡ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭■❋❚✮ ❛❧❧ ♦❢ t❤❡s❡ ❛r❡ ❡①❛♠♣❧❡s ♦❢ ❧✐♥❡❛r ♦♣❡r❛t♦rs✱ A [v] (①)

A✱ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r v✱ t❤❡ ❢✉♥❝t✐♦♥ t♦ ✇❤✐❝❤ ✐t ✐s ❛♣♣❧✐❡❞ ①✱ t❤❡ ❢✉♥❝t✐♦♥✬s ❛r❣✉♠❡♥t

q✳✈✳ ▼❡✉❝❝✐✱ ❆♣♣❡♥❞✐① ❇✳✸ ✭♥✳❜✳ fX ❡①✐sts ✐✛ FX ✐s ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s❀ φX ❛❧✇❛②s ❡①✐sts✮

✷✸ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥

▲❡❝t✉r❡ ✶ ❡①❡r❝✐s❡s

▼❡✉❝❝✐ ❡①❡r❝✐s❡s

♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✶✳✶✳✶✱ ✶✳✶✳✷✱ ✶✳✶✳✸✱ ✶✳✶✳✺✱ ✶✳✶✳✻✱ ✶✳✷✳✷ ▼❆❚▲❆❇✿ ✶✳✶✳✹✱ ✶✳✶✳✼✱ ✶✳✶✳✽

♣r♦❥❡❝t

♣✐❝❦ ❛ ❝♦✉♥tr② ❢♦r ②♦✉r ♣r♦❥❡❝t ✭❡♥s✉r✐♥❣ t❤❛t ♥♦ ♦♥❡ ❡❧s❡ ✐s ✉s✐♥❣ ✐t✮✱ ❛♥❞ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ❡q✉✐t② ✐♥❞❡① ❛♥❞ ❝♦♠♠♦❞✐t② ✭✐❞❡❛❧❧② ❣♦❧❞✮✳ ❇❡❣✐♥ t♦ ❡①♣❡r✐♠❡♥t ✇✐t❤ ②♦✉r ■♥t❡r❛❝t✐✈❡ ❇r♦❦❡rs tr❛❞✐♥❣ ♣❧❛t❢♦r♠✳

✷✹ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙✉♠♠❛r② st❛t✐st✐❝s

❑❡② s✉♠♠❛r② ♣❛r❛♠❡t❡rs

❢✉❧❧ ❞✐str✐❜✉t✐♦♥s ❝❛♥ ❜❡ ❡①♣❡♥s✐✈❡ t♦ r❡♣r❡s❡♥t ✇❤❛t s✉♠♠❛r② ✐♥❢♦r♠❛t✐♦♥ ❤❡❧♣s ❝❛♣t✉r❡ ❦❡② ❢❡❛t✉r❡s❄

✶ ❧♦❝❛t✐♦♥✱ Loc {X}

✐❢ ❤❛❞ ♦♥❡ ❣✉❡ss ❛s t♦ ✇❤❡r❡ X ✇♦✉❧❞ t❛❦❡ ✐ts ✈❛❧✉❡ s❤♦✉❧❞ s❛t✐s❢② Loc {a} = a ❛♥❞ ❛✣♥❡ ❡q✉✐✈❛r✐❛♥❝❡ Loc {a + bX} = a + bLoc {X} t♦ ❡♥s✉r❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ♠❡❛s✉r❡♠❡♥t s❝❛❧❡✴❝♦♦r❞✐♥❛t❡ s②st❡♠

✷ ❞✐s♣❡rs✐♦♥✱ Dis {X}

❤♦✇ ❛❝❝✉r❛t❡ t❤❡ ❧♦❝❛t✐♦♥ ❣✉❡ss✱ ❛❜♦✈❡✱ ✐s ❛✣♥❡ ❡q✉✐✈❛r✐❛♥❝❡ ✐s ♥♦✇ Dis {a + bX} = |b| Dis {X} ✇❤❡r❡ |·| ❞❡♥♦t❡s ❛❜s♦❧✉t❡ ✈❛❧✉❡

✸ ③✲s❝♦r❡ ♥♦r♠❛❧✐s❡s ❛ r✈✱ ZX ≡ X−Loc{X}

Dis{X} ✿ ✵ ❧♦❝❛t✐♦♥❀ ✶ ❞✐s♣❡rs✐♦♥

❛✣♥❡ ❡q✉✐✈❛r✐❛♥❝❡ ♦❢ ❧♦❝❛t✐♦♥ ✫ ❞✐s♣❡rs✐♦♥ ⇔ (Za+bX)✷ = (ZX)✷

✷✻ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙✉♠♠❛r② st❛t✐st✐❝s

▼♦st ❝♦♠♠♦♥ ❧♦❝❛t✐♦♥ ❛♥❞ ❞✐s♣❡rs✐♦♥ ♠❡❛s✉r❡s

❵❧♦❝❛❧✬ ❵s❡♠✐✲❧♦❝❛❧✬ ❵❣❧♦❜❛❧✬ ❧♦❝❛t✐♦♥ ♠♦❞❡✱ Mod {X} ♠❡❞✐❛♥✱ Med {X} ♠❡❛♥ ✴ ❡①♣✬❞ ✈❛❧✱ E {X} ❛r❣♠❛①x∈R fX (x) Med{X}

−∞

fX (x) dx = ✶

✷

∞

−∞ xfX (x) dx

❞✐s♣❡rs✐♦♥ ♠♦❞❛❧ ❞✐s♣❡rs✐♦♥ ✐♥t❡rq✉❛♥t✐❧❡ r❛♥❣❡ ✈❛r✐❛♥❝❡ ∞

−∞ (x − E {X})✷ fX (x) dx

❵❣❧♦❜❛❧✬ ♠❡❛s✉r❡s ❛r❡ ❢♦r♠❡❞ ❢r♦♠ t❤❡ ✇❤♦❧❡ ❞✐str✐❜✉t✐♦♥ ❵s❡♠✐✲❧♦❝❛❧✬ ♠❡❛s✉r❡s ❛r❡ ❢♦r♠❡❞ ❢r♦♠ ❤❛❧❢ ✭♦r s♦✮ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ❵❧♦❝❛❧✬ ♠❡❛s✉r❡s ❛r❡ ❞r✐✈❡♥ ❜② ✐♥❞✐✈✐❞✉❛❧ ♦❜s❡r✈❛t✐♦♥s ❣❡♥❡r❛❧❧②✱ ✇❡ ❞❡✜♥❡ Dis {X} ≡ X − Loc {X} X;p ✇❤❡r❡ gX;p ≡ (E {|g (X)|p})

✶ p ✐s t❤❡ ♥♦r♠ ♦♥ t❤❡ ✈❡❝t♦r s♣❛❝❡ Lp

X

p = ✶ ✐s t❤❡ ♠❡❛♥ ❛❜s♦❧✉t❡ ❞❡✈✐❛t✐♦♥✱ MAD {X} ≡ E {|X − E {X}|} p = ✷ ✐s t❤❡ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥✱ Sd {X} ≡

• E
• |X − E {X}|✷ ✶

✷

✷✼ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙✉♠♠❛r② st❛t✐st✐❝s

❍✐❣❤❡r ♦r❞❡r ♠♦♠❡♥ts

✶ kt❤✲r❛✇ ♠♦♠❡♥t

RMX

k ≡ E

• X k

✐s t❤❡ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ kt❤ ♣♦✇❡r ♦❢ X

✷ kt❤✲❝❡♥tr❛❧ ♠♦♠❡♥t ✐s ♠♦r❡ ❝♦♠♠♦♥❧② ✉s❡❞

CMX

k ≡ E

• (X − E {X})k

❞❡✲♠❡❛♥s t❤❡ r❛✇ ♠♦♠❡♥t✱ ♠❛❦✐♥❣ ✐t ❧♦❝❛t✐♦♥✲✐♥❞❡♣❡♥❞❡♥t

s❦❡✇♥❡ss✱ ❛ ♠❡❛s✉r❡ ♦❢ s②♠♠❡tr②✱ ✐s t❤❡ ♥♦r♠❛❧✐s❡❞ ✸r❞ ❝❡♥tr❛❧ ♠♦♠❡♥t Sk {X} ≡ CMX

✸

(Sd {X})✸ ❦✉rt♦s✐s ♠❡❛s✉r❡s t❤❡ ✇❡✐❣❤ts ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥✬s t❛✐❧ r❡❧❛t✐✈❡ t♦ ✐ts ❝❡♥tr❡ Ku {X} ≡ CMX

✹

(Sd {X})✹

✷✽ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

❯♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥✿ X ∼ U ([a, b])

s✐♠♣❧❡st ❞✐str✐❜✉t✐♦♥❀ s❤❛❧❧ ❜❡ ✉s❡❢✉❧ ✇❤❡♥ ♠♦❞❡❧❧✐♥❣ ❝♦♣✉❧❛s ❢✉❧❧② ❞❡s❝r✐❜❡❞ ❜② t✇♦ ♣❛r❛♠❡t❡rs✱ a ✭❧♦✇❡r ❜♦✉♥❞✮ ❛♥❞ b ✭✉♣♣❡r ❜♦✉♥❞✮ ❛♥② ♦✉t❝♦♠❡ ✐♥ t❤❡ [a, b] ✐s ❡q✉❛❧❧② ❧✐❦❡❧② ❝❧♦s❡❞ ❢♦r♠ r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r f U

a,b (x) , F U a,b (x) , φU a,b (ω) ❛♥❞ QU a,b (p)

st❛♥❞❛r❞ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ✐s U ([✵, ✶])

✸✵ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

◆♦r♠❛❧ ✭●❛✉ss✐❛♥✮ ❞✐str✐❜✉t✐♦♥✿ X ∼ N

• µ, σ✷

♠♦st ✇✐❞❡❧② ✉s❡❞✱ st✉❞✐❡❞ ❞✐str✐❜✉t✐♦♥ ❢✉❧❧② ❞❡s❝r✐❜❡❞ ❜② t✇♦ ♣❛r❛♠❡t❡rs✱ µ ✭♠❡❛♥✮ ❛♥❞ σ✷ ✭✈❛r✐❛♥❝❡✮ st❛♥❞❛r❞ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ µ = ✵ ❛♥❞ σ✷ = ✶ ❛s ❛ st❛❜❧❡ ❞✐str✐❜✉t✐♦♥✱ t❤❡ s✉♠s ♦❢ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ r✈✬s ❛r❡ ♥♦r♠❛❧ ❝❧♦s❡❞ ❢♦r♠ r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r f N

µ,σ✷ (x) , F N µ,σ✷ (x) , φN µ,σ✷ (ω) ❛♥❞

QN

µ,σ✷ (p)

✇❤② ❞♦ ✇❡ ❝❛r❡ t❤❛t Ku {X} = ✸❄

✸✶ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

❈❛✉❝❤② ❞✐str✐❜✉t✐♦♥✿ X ∼ Ca

• µ, σ✷

❵❢❛t✲t❛✐❧❡❞✬ ❞✐str✐❜✉t✐♦♥✿ ✇❤❡♥ ♠✐❣❤t t❤✐s ❜❡ ✉s❡❢✉❧❄ ❢✉❧❧② ❞❡s❝r✐❜❡❞ ❜② t✇♦ ♣❛r❛♠❡t❡rs✱ µ ❛♥❞ σ✷ f Ca

µ,σ✷ (x) ≡

✶ π √ σ✷

• ✶ + (x − µ)✷

σ✷ −✶ ✇❤❛t ❛r❡ E {X} , Var {X} , Sk {X} ❛♥❞ Ku {X}❄

s❡❡ ❤❡r❡ ❢♦r ❛ ❞✐s❝✉ss✐♦♥

st❛♥❞❛r❞ ❈❛✉❝❤② ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ µ = ✵ ❛♥❞ σ✷ = ✶

✭❋❨■✿ ✐❢ X, Y ∼ NID (✵, ✶) t❤❡♥ X

Y ∼ Ca (✵, ✶)✮

✸✷ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

❙t✉❞❡♥t t ❞✐str✐❜✉t✐♦♥✿ X ∼ St

• ν, µ, σ✷

❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ♣❛r❛♠❡t❡r✱ ν✱ ❞❡t❡r♠✐♥❡s ❢❛t♥❡ss ♦❢ t❛✐❧s ❛♥❛❧②t✐❝❛❧ ❡①♣r❡ss✐♦♥s ❢♦r f St

ν,µ,σ✷, F St ν,µ,σ✷ ❛♥❞ φSt ν,µ,σ✷ ✉s❡

t❤❡ ❣❛♠♠❛✱ ❜❡t❛ ❛♥❞ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s❀ ♥♦♥❡ ❢♦r QSt

ν,µ,σ✷

❧✐♠✐t ♦❢ ❛♥❛❧②t✐❝❛❧ ❡①♣r❡ss✐♦♥s q✉✐❝❦❧② r❡❛❝❤❡❞

st❛♥❞❛r❞ ❙t✉❞❡♥t ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ µ = ✵ ❛♥❞ σ✷ = ✶ ✇❤❡♥ ❛r❡ E {X} , Var {X} , Sk {X} ❛♥❞ Ku {X} ❞❡✜♥❡❞❄

❊①❛♠♣❧❡ ✭ν = ✸✮

ν → ∞ ⇒ St

• ν, µ, σ✷

→d N

• µ, σ✷

ν → ✶ ⇒ St

• ν, µ, σ✷

→d Ca

• µ, σ✷

−✸ −✷ −✶ ✵ ✶ ✷ ✸ ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ① ②

✸✸ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

▲♦❣✲♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✿ X ∼ LogN

• µ, σ✷

✐❢ Y ∼ N

• µ, σ✷

t❤❡♥ X ≡ eY ∼ LogN

• µ, σ✷

✭❇❛✐❧❡②✿ s❤♦✉❧❞ ❜❡ ❝❛❧❧❡❞ ❵❡①♣✲♥♦r♠❛❧✬ ❞✐str✐❜✉t✐♦♥❄✮ ♥♦✇ φLogN

µ,σ✷ ❤❛s ♥♦ ❦♥♦✇♥

❛♥❛❧②t✐❝ ❢♦r♠ ♣r♦♣❡rt✐❡s

X > ✵ (% ❝❤❛♥❣❡s ✐♥ X) ∼ N ❛s②♠♠❡tr✐❝ ✭♣♦s✐t✐✈❡❧② s❦❡✇❡❞✮

❝♦♠♠♦♥❧② ❛♣♣❧✐❡❞ t♦ st♦❝❦ ♣r✐❝❡s ✭❍✉❧❧ ✭✷✵✵✾✱ ➓✶✷✳✻✱ ➓✶✸✳✶✮✱ ❙t❡❢❛♥✐❝❛ ✭✷✵✶✶✱ ➓✹✳✻✮✮

❊①❛♠♣❧❡

−✸ −✷ −✶ ✵ ✶ ✷ ✸ ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ✵✳✺ ✵✳✻ ① ②

✸✹ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

• ❛♠♠❛ ❞✐str✐❜✉t✐♦♥✿ X ∼ Ga
• ν, µ, σ✷

❧❡t Y✶, . . . , Yν ∼ IID s✳t✳ Yt ∼ N

• µ, σ✷

∀t ∈ {✶, . . . ν} ♥♦♥✲❝❡♥tr❛❧ ❣❛♠♠❛ ❞✐str✐❜✉t✐♦♥✱ X ≡ ν

t=✶ Y ✷ t ∼ Ga

• ν, µ, σ✷

ν✿ ❞❡❣r❡❡s✲♦❢✲❢r❡❡❞♦♠ ✭s❤❛♣❡✮❀ µ✿ ♥♦♥✲❝❡♥tr❛❧✐t②❀ σ✷✿ s❝❛❧❡ ❇❛②❡s✐❛♥s✿ ❡❛❝❤ ♦❜s❡r✈❛t✐♦♥ ✐s ❛♥ r✈ ⇒ t❤❡✐r ✈❛r✐❛♥❝❡ ∼ Ga

✶ µ = ✵ ⇒ ❝❡♥tr❛❧ ❣❛♠♠❛

❞✐str✐❜✉t✐♦♥✱ X ∼ Ga

• ν, σ✷

✭♠♦st ❝♦♠♠♦♥✮

✷ σ✷ = ✶ ⇒ ♥♦♥✲❝❡♥tr❛❧ ❝❤✐✲sq✉❛r❡

❞✐str✐❜✉t✐♦♥

✸ µ = ✵, σ✷ = ✶ ⇒ ❝❤✐✲sq✉❛r❡

❞✐str✐❜✉t✐♦♥✱ X ∼ χ✷

ν

❊①❛♠♣❧❡ ✭µ = ✵, σ✷ = ✶✮

−✸ −✷ −✶ ✵ ✶ ✷ ✸ ✵✳✵✵ ✵✳✵✺ ✵✳✶✵ ✵✳✶✺ ✵✳✷✵ ✵✳✷✺ ① ②

❳ ∼ W ✸✺ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

❊♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥✿ X ∼ Em (iT)

❞❛t❛ ❞❡✜♥❡s ❞✐str✐❜✉t✐♦♥✿ ❢✉t✉r❡ ♦❝❝✉rs ✇✐t❤ s❛♠❡ ♣r♦❜❛❜✐❧✐t② ❛s ♣❛st fiT (x) ≡ ✶ T

T

• t=✶

δ(xt) (x) FiT (x) ≡ ✶ T

T

• t=✶

H(xt) (x)

δ(xt) (·) ✐s ❉✐r❛❝✬s ❞❡❧t❛ ❢✉♥❝t✐♦♥ ❝❡♥tr❡❞ ❛t xt✱ ❛ ❣❡♥❡r❛❧✐s❡❞ ❢✉♥❝t✐♦♥ ✭✐❢ ✇✐s❤ t♦ tr❡❛t X ❛s ❞✐s❝r❡t❡✱ ❑r♦♥❡❝❦❡r✬s ❞❡❧t❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡s ♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥✮ H(xt) (·) ✐s ❍❡❛✈✐s✐❞❡✬s st❡♣ ❢✉♥❝t✐♦♥✱ ✇✐t❤ ✐ts st❡♣ ❛t xt

✇❤❛t ❞♦ t❤❡s❡ ❧♦♦❦ ❧✐❦❡❄ ❲❤❛t ❞♦ r❡❣✉❧❛r✐s❡❞ ✈❡rs✐♦♥s ❧♦♦❦ ❧✐❦❡❄ ❞❡✜♥✐♥❣ QiT (p) ♦❜t❛✐♥❡❞ ❜② ❜❛♥❞✇✐❞t❤ t❡❝❤♥✐q✉❡s ♦❢ ❆♣♣❡♥❞✐① ❇✿ ♦r❞❡r ♦❜s❡r✈❛t✐♦♥s✱ t❤❡♥ ❝♦✉♥t ❢r♦♠ ❧♦✇❡st

❳ ∼ Em ✸✻ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

▲❡❝t✉r❡ ✷ ❡①❡r❝✐s❡s

▼❡✉❝❝✐ ❡①❡r❝✐s❡s

♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✶✳✷✳✺ ✭♥♦t ▼❆❚▲❆❇✮✱ ✶✳✷✳✻✱ ✶✳✷✳✼ ▼❆❚▲❆❇✿ ✶✳✷✳✸✱ ✶✳✷✳✺ ✭▼❆❚▲❆❇✮

♣r♦❥❡❝t

❡①♣❧♦r❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ r❡t✉r♥s ❢♦r ②♦✉r ❡q✉✐t② ✐♥❞❡①✱ ②♦✉r ❝♦♠♠♦❞✐t② ❛♥❞ ②♦✉r ❝✉rr❡♥❝✐❡s✬ ❡①❝❤❛♥❣❡ r❛t❡ ✈ t❤❡ ❯❙❉

✸✼ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s

❉✐r❡❝t ❡①t❡♥s✐♦♥s ♦❢ ✉♥✐✈❛r✐❛t❡ st❛t✐st✐❝s

✐❢ ✐♥t❡r❡st❡❞ ✐♥ ♣♦rt❢♦❧✐♦s ✭♦r ❡✈❡♥ ❛r❜✐tr❛❣❡✮✱ ♠✉st ❜❡ ❛❜❧❡ t♦ ❝♦♥s✐❞❡r ❤♦✇ ❛♥ ❛ss❡t✬s ♠♦✈❡♠❡♥ts ❞❡♣❡♥❞ ♦♥ ♦t❤❡rs✬ N✲❞✐♠❡♥s✐♦♥❛❧ r✈✱ ❳ ≡ (X✶, . . . , XN)′✱ s♦ t❤❛t ① ∈ RN ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ P {❳ ∈ R} ≡

• R

f❳ (①) d①, st f❳ (①) ≥ ✵,

• RN f❳ (①) d① = ✶

❝✉♠✉❧❛t✐✈❡ ♦r ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✭❞❢✱ ❉❋✱ ❈❉❋✱ ❏❉❋ ✳ ✳ ✳ ✮ F❳ (①) ≡ P {❳ ≤ ①} = x✶

−∞

· · · xN

−∞

f❳ (u✶, . . . , uN) duN · · · du✶ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ φ❳ (ω) ≡ E

• eiω′❳

, ω ∈ RN ✇❤❛t ❛❜♦✉t t❤❡ q✉❛♥t✐❧❡❄ ✭❤✐♥t✿ F❳ : RN → R✶✮

✸✾ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s

▼❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥✴❞❡♥s✐t② ♦❢ ❳B

♣❛rt✐t✐♦♥ ❳ ✐♥t♦ K✲❞✐♠❡♥s✐♦♥❛❧ ❳A ❛♥❞ (N − K)✲❞✐♠❡♥s✐♦♥❛❧ ❳B ❞✐str✐❜✉t✐♦♥ ♦❢ ❳B ✇❤❛t❡✈❡r ❳A✬s ✭t❡❝❤♥✐❝❛❧❧②✿ ✐♥t❡❣r❛t❡s ♦✉t ❳A✮ F❳B (①B) ≡P {❳B ≤ ①B} =P {❳A ≤ ∞, ❳B ≤ ①B} =F❳ (∞, ①B) f❳B (①B) ≡

• RK f❳ (①A, ①B) d①A

φ❳B (ω) ≡E

• eiω′❳B
• =E
• eiψ′❳A+iω′❳B
• |ψ=✵

=φ❳ (✵, ω)

❊①❛♠♣❧❡

① ❴ ❜ ①❴❛ ❢

✹✵ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s x y z

❲❤❛t✱ r♦✉❣❤❧②✱ ❞♦ t❤❡ ♠❛r❣✐♥❛❧s ♦❢ t❤✐s ♣❞❢ ❧♦♦❦ ❧✐❦❡❄

❝♦♣✉❧❛ ❞❡✜♥❡❞ ✹✶ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s

❈♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥✴❞❡♥s✐t② ♦❢ ❳A ❣✐✈❡♥ ①B

❡✳❣✳ ✜① ❛ss❡ts B✬s r❡t✉r♥s ❛t ①B❀ ✇❤❛t ✐s t❤❛t ♦❢ ❛ss❡ts A❄ f❳A|①B (①A) ≡ f❳ (①A, ①B) f❳B (①B) ❝❛♥ ❞❡❝♦♠♣♦s❡ ❏❉❋ ✐♥t♦ ♣r♦❞✉❝t ♦❢ ♠❛r❣✐♥❛❧ ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❇❛②❡s✬ r✉❧❡ ❢♦r ✉♣❞❛t✐♥❣ ❜❡❧✐❡❢s ✐s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ f❳A|①B (①A) = f❳B|①A (①B) f❳A (①A) f❳B (①B)

❊①❛♠♣❧❡

①❴❜ ①❴❛ ❢

✹✷ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s

▲♦❝❛t✐♦♥ ♣❛r❛♠❡t❡r✱ Loc {❳}

❞❡s✐❞❡r❛t❛ ♦❢ ❧♦❝❛t✐♦♥ ❡①t❡♥❞ ❞✐r❡❝t❧② ❢r♦♠ ✉♥✐✈❛r✐❛t❡ ❝❛s❡

✶

❢♦r ❝♦♥st❛♥t ♠✱ Loc {♠} = ♠

✷

❢♦r ✐♥✈❡rt✐❜❧❡ ❇✱ ❛✣♥❡ ❡q✉✐✈❛r✐❛♥❝❡ ♥♦✇ Loc {❛ + ❇❳} = ❛ + ❇Loc {❳}

❡①♣❡❝t❡❞ ✈❛❧✉❡

✶

E {❳} = (E {X✶} , . . . , E {XN})′

✷

❛✣♥❡ ❡q✉✐✈❛r✐❛♥❝❡ ♣r♦♣❡rt② ❤♦❧❞s ❢♦r ❛♥② ❝♦♥❢♦r♠❛❜❧❡ ❇✱ ♥♦t ❥✉st ✐♥✈❡rt✐❜❧❡ ✭Med {❳} , Mod {❳} r❡q✉✐r❡ ✐♥✈❡rt✐❜❧❡✮

✸

r❡❧❛t✐✈❡❧② ❡❛s② t♦ ❝❛❧❝✉❧❛t❡ ✇❤❡♥ φ❳ ❦♥♦✇♥✱ ❛♥❛❧②t✐❝❛❧ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ➓❚✷✳✶✵✮

✹✸ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s

❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r✱ Dis {❳}

r❡❝❛❧❧✿ ✐♥ t❤❡ ✉♥✐✈❛r✐❛t❡ ❝❛s❡✱ t❤❡ z✲s❝♦r❡ ♥♦r♠❛❧✐s❡s ❛ ❞✐str✐❜✉t✐♦♥ s♦ t❤❛t ✐t ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r ❛✣♥❡ tr❛♥s❢♦r♠❛t✐♦♥s |Za+bX| = |ZX| ≡

• (X − Loc {X}) (X − Loc {X})

Dis {X} ❧❡t Σ ❜❡ ❛ s②♠♠❡tr✐❝ P❉ ♦r P❙❉ ♠❛tr✐①❀ t❤❡♥ ▼❛❤❛❧❛♥♦❜✐s ❞✐st❛♥❝❡ ❢r♦♠ ① t♦ µ✱ ♥♦r♠❛❧✐s❡❞ ❜② t❤❡ ♠❡tr✐❝ Σ✱ ✐s Ma (①, µ, Σ) ≡

• (① − µ)′ Σ−✶ (① − µ)

❣✐✈❡♥ ❛♥ ❡❧❧✐♣s♦✐❞ ❝❡♥tr❡❞ ❛t µ ✇❤♦s❡ ♣r✐♥❝✐♣❛❧ ❛①❡s✬ ❧❡♥❣t❤s ❡q✉❛❧ t❤❡ sq✉❛r❡ r♦♦ts ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ Σ✱ ❛❧❧ ① ♦♥ ✐ts s✉r❢❛❝❡ ❤❛✈❡ t❤❡ s❛♠❡ ▼❛❤❛❧❛♥♦❜✐s ❞✐st❛♥❝❡ ❢r♦♠ µ

■■❉ ❤❡✉r✐st✐❝ t❡st ✷

♠✉❧t✐✈❛r✐❛t❡ z✲s❝♦r❡ ✐s t❤❡♥ Ma❳ ≡ Ma (❳, Loc {❳} , DisSq {❳}) ❜❡♥❝❤♠❛r❦ ✭sq✉❛r❡❞✮ ❞✐s♣❡rs✐♦♥ ♦r s❝❛tt❡r ♣❛r❛♠❡t❡r✿ ❝♦✈❛r✐❛♥❝❡

✹✹ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❉❡♣❡♥❞❡♥❝❡

❈♦rr❡❧❛t✐♦♥

♥♦r♠❛❧✐s❡❞ ❝♦✈❛r✐❛♥❝❡ ρ (Xm, Xn) = Cor {Xm, Xn} ≡ Cov {Xm, Xn} Sd {Xm} Sd {Xn} ∈ [−✶, ✶] ✇❤❡r❡ Cov {Xm, Xn} ≡ E {(Xm − E {Xm}) (Xn − E {Xn})} ✇❤❡♥ ✐s t❤✐s ♥♦t ❞❡✜♥❡❞❄ ❛ ♠❡❛s✉r❡ ♦❢ ❧✐♥❡❛r ❞❡♣❡♥❞❡♥❝❡✱ ✐♥✈❛r✐❛♥t ✉♥❞❡r str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s ρ (αm + βmXm, αn + βnXn) = ρ (Xm, Xn) ❢❛❧❧❛❝② ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✷✹✶✮✿ ❣✐✈❡♥ ♠❛r❣✐♥❛❧ ❞❢s F✶ ❛♥❞ F✷ ❛♥❞ ❛♥② ρ ∈ [−✶, ✶]✱ ❝❛♥ ❛❧✇❛②s ✜♥❞ ❛ ❏❉❋ F ❜✐♥❞✐♥❣ t❤❡♠ t♦❣❡t❤❡r

tr✉❡ ❢♦r ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s❀ ❣❡♥❡r❛❧❧②✱ ❛tt❛✐♥❛❜❧❡ ❝♦rr❡❧❛t✐♦♥s ❛r❡ ❛ str✐❝t s✉❜s❡t ♦❢ [−✶, ✶] ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ❊①❛♠♣❧❡ ✼✳✷✾✮

✹✻ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❉❡♣❡♥❞❡♥❝❡

❙t❛♥❞❛r❞ ♥♦r♠❛❧ ♠❛r❣✐♥❛❧s✱ ρ ≈ .✼

❢❛❧❧❛❝② ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✷✸✾✮✿ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s ❛♥❞ ♣❛✐r✇✐s❡ ❝♦rr❡❧❛t✐♦♥s ♦❢ ❛ r✈ ❞❡t❡r♠✐♥❡ ✐ts ❥♦✐♥t ❞✐str✐❜✉t✐♦♥

✹✼ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❉❡♣❡♥❞❡♥❝❡

■♥❞❡♣❡♥❞❡♥❝❡

✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ♦♥❡ ✈❛r✐❛❜❧❡ ❞♦❡s ♥♦t ❛✛❡❝t ❞✐str✐❜✉t✐♦♥ ♦❢ ♦t❤❡rs f❳B (①B) = f❳B|①A (①B) ♣r♦❜❛❜✐❧✐t② ♦❢ t✇♦ ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts✿ P {e ∩ f} = P {e} P {f} F❳ (①A, ①B) = F❳A (①A) F❳B (①B) ❢r♦♠ ❞❡✜♥✐t✐♦♥s ♦❢ ❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥❝❡ ✭tr② ✐t✦✮ f❳ (①A, ①B) = f❳A (①A) f❳B (①B) ❛❜♦✈❡ tr✉❡ ✐❢ ❳A, ❳B tr❛♥s❢♦r♠❡❞ ❜② ❛r❜✐tr❛r② g (·) ❛♥❞ h (·)✿ ✐❢ ①A ❞♦❡s♥✬t ❡①♣❧❛✐♥ ❳B✱ tr❛♥s❢♦r♠❡❞ ✈❡rs✐♦♥s ✇♦♥✬t ❡✐t❤❡r

❧✐♥❡❛r r❡t✉r♥s ♣❧♦t

t❤❡r❡❢♦r❡ ❛❧❧♦✇s ♥♦♥✲❧✐♥❡❛r r❡❧❛t✐♦♥s ✐♥❞❡♣❡♥❞❡♥t ✐♠♣❧✐❡s ✉♥❝♦rr❡❧❛t❡❞✱ ❜✉t ♥♦t t❤❡ ❝♦♥✈❡rs❡

❊①❛♠♣❧❡

• ✐✈❡♥ X ✷ + Y ✷ = ✶✱ ❛r❡ t❤❡ r✈s X ❛♥❞ Y ✭✉♥✮❝♦rr❡❧❛t❡❞✱ ✭✐♥✮❞❡♣❡♥❞❡♥t❄

❍✐♥t✿ ✐❢ ✜tt✐♥❣ yi = mxi + b + εi✱ ✇❤❛t ❛r❡ m, ˆ m❄

✹✽ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

❯♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥

✐❞❡❛ ✐s ❛s ✐♥ ✉♥✐✈❛r✐❛t❡ ❝❛s❡✱ ❜✉t ❞♦♠❛✐♥ ♠❛② ❜❡ ❛♥②t❤✐♥❣ ♦❢t❡♥ ❡❧❧✐♣t✐❝❛❧ ❞♦♠❛✐♥✱ Eµ,Σ ✇❤❡r❡ µ ✐s ❝❡♥tr♦✐❞✱ Σ ✐s ♣♦s✐t✐✈❡ ♠❛tr✐①

❊①❛♠♣❧❡

fX✶,X✷ (x✶, x✷) = ✶ πI{x✷

✶ +x✷ ✷ ≤✶} (x✶, x✷)

✇❤❡r❡ IS ✐s t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦♥ t❤❡ s❡t S ♠❛r❣✐♥❛❧ ❞❡♥s✐t②✿ fX✶ (x✶) = √

✶−x✷

✶

−√ ✶−x✷

✶

✶ πdx✷ = ✷ π

• ✶ − x✷

✶

❝♦♥❞✐t✐♦♥❛❧ ❞❡♥s✐t②✿ fX✶|x✷ (x✶) =

fX✶,X✷(x✶,x✷) fX✷(x✷)

=

✶ ✷√ ✶−x✷

✷

❛r❡ X✶ ❛♥❞ X✷ ✭✉♥✮❝♦rr❡❧❛t❡❞✱ ✭✐♥✮❞❡♣❡♥❞❡♥t❄

✺✵ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

◆♦r♠❛❧ ✭●❛✉ss✐❛♥✮ ❞✐str✐❜✉t✐♦♥✿ ❳ ∼ N (µ, Σ)

♠♦st ✇✐❞❡❧② ✉s❡❞✱ st✉❞✐❡❞ ❞✐str✐❜✉t✐♦♥ ❢✉❧❧② ❞❡s❝r✐❜❡❞ ❜② t✇♦ ♣❛r❛♠❡t❡rs✱ µ ✭❧♦❝❛t✐♦♥✮ ❛♥❞ Σ ✭❞✐s♣❡rs✐♦♥✮ st❛♥❞❛r❞ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ µ = ✵ ❛♥❞ Σ = ■ ✭✐❞❡♥t✐t② ♠❛tr✐①✮ ❝❧♦s❡❞ ❢♦r♠ r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r f N

µ,Σ (①) , F N µ,Σ (①)✱ ❛♥❞ φN µ,Σ (ω)

❛s s②♠♠❡tr✐❝ ❛♥❞ ✉♥✐♠♦❞❛❧ E {❳} = Mod {❳} = Med {❳} = µ Cov {❳} = Σ ♠❛r❣✐♥❛❧✱ ❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥s ❛❧s♦ ♥♦r♠❛❧

✺✶ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

❙t✉❞❡♥t t ❞✐str✐❜✉t✐♦♥✿ ❳ ∼ St (ν, µ, Σ)

❛❣❛✐♥✱ s②♠♠❡tr✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛❜♦✉t ❛ ♣❡❛❦ ❛❣❛✐♥✱ t❤r❡❡ ♣❛r❛♠❡t❡rs

❛s s②♠♠❡tr✐❝ ❛♥❞ ✉♥✐♠♦❞❛❧✱ E {❳} = Mod {❳} = Med {❳} = µ s❝❛tt❡r ♣❛r❛♠❡t❡r = ❝♦✈❛r✐❛♥❝❡✿ Cov {❳} =

ν ν−✷Σ

st❛♥❞❛r❞ ❙t✉❞❡♥t t ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ µ = ✵ ❛♥❞ Σ = ■ ❢♦r♠ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ❞❡♣❡♥❞s ♦♥ ✇❤❡t❤❡r ν ❡✈❡♥ ♦r ♦❞❞❀ s❡❡ ❇❡r❣ ❛♥❞ ❱✐❣♥❛t ✭✷✵✵✽✮ ❢♦r ❛ ❞✐✛❡r❡♥t ✭❝♦♥tr❛r②❄✮ ❛♣♣r♦❛❝❤ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s ❛r❡ ❛❧s♦ t❀ ❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥s ❛r❡ ♥♦t❀ t❤✉s✱ ✐❢ ❳ ∼ St✱ ❝❛♥✬t ❜❡ ✐♥❞❡♣❡♥❞❡♥t

✺✷ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

❈❛✉❝❤② ❞✐str✐❜✉t✐♦♥✿ ❳ ∼ Ca (µ, Σ)

❛s ✐♥ t❤❡ ✉♥✐✈❛r✐❛t❡ ❝❛s❡✱ t❤❡ ❢❛t✲t❛✐❧❡❞ ❧✐♠✐t ♦❢ t❤❡ ❙t✉❞❡♥t t✲❞✐str✐❜✉t✐♦♥✿ Ca (µ, Σ) = St (✶, µ, Σ) st❛♥❞❛r❞ ❈❛✉❝❤② ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ µ = ✵ ❛♥❞ Σ = ■ ✭✐❞❡♥t✐t② ♠❛tr✐①✮ s❛♠❡ ♣r♦❜❧❡♠ ✇✐t❤ ♠♦♠❡♥ts ❛s ✉♥✐✈❛r✐❛t❡ ❝❛s❡

✺✸ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

▲♦❣✲❞✐str✐❜✉t✐♦♥s

❡①♣♦♥❡♥t✐❛❧s ♦❢ ♦t❤❡r ❞✐str✐❜✉t✐♦♥s✱ ❛♣♣❧✐❡❞ ❝♦♠♣♦♥❡♥t✲✇✐s❡ t❤✉s✱ ✉s❡❢✉❧ ❢♦r ♠♦❞❡❧❧✐♥❣ ♣♦s✐t✐✈❡ ✈❛❧✉❡s ✐❢ ❨ ❤❛s ♣❞❢ f❨ t❤❡♥ ❳ ≡ e❨ ✐s ❧♦❣✲❨ ❞✐str✐❜✉t❡❞

❊①❛♠♣❧❡ ✭▲♦❣✲♥♦r♠❛❧✮

▲❡t ❨ ∼ N (µ, Σ)✳ ❚❤❡♥✱ ✐❢ ❳ ≡ e❨ ✱ s♦ t❤❛t Xi ≡ eYi ❢♦r ❛❧❧ i = ✶, . . . , N✱ ❳ ∼ LogN (µ, Σ)✳

✺✹ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

❲✐s❤❛rt ❞✐str✐❜✉t✐♦♥✿ ❲ ∼ W (ν, Σ)

❝♦♥s✐❞❡r N✲❞✐♠❡♥s✐♦♥❛❧s ■■❉ r✈s ❳t ∼ N (✵, Σ) ❢♦r t = ✶, . . . , ν ≥ N t❤❡♥ ❲✐s❤❛rt ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ν ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ✐s t❤❡ r❛♥❞♦♠ ♠❛tr✐① ❲ ≡ ❳✶❳ ′

✶ + · · · + ❳ν❳ ′ ν

❛s Σ ✐s s②♠♠❡tr✐❝ ❛♥❞ P❉✱ s♦ ✐s ❲ ♠✉❧t✐✈❛r✐❛t❡ ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ❣❛♠♠❛ ❞✐str✐❜✉t✐♦♥

X ∼ Ga

❢✉rt❤❡r♠♦r❡✱ ❣✐✈❡♥ ❣❡♥❡r✐❝ ❛✱ ❲ ∼ W (ν, Σ) ⇒ ❛′❲❛ ∼ Ga

• ν, ❛′Σ❛
• ❛s ✐♥✈❡rs❡ ♦❢ s②♠♠❡tr✐❝✱ P❉ ♠❛tr✐① ✐s s②♠♠❡tr✐❝✱ P❉✱ ✐♥✈❡rs❡ ❲✐s❤❛rt

❩ −✶ ∼ W

• ν, Ψ−✶

⇒ ❩ ∼ IW (ν, Ψ) ❛s ❛ r❛♥❞♦♠ P❉ ♠❛tr✐①✱ ❲✐s❤❛rt ✉s❡❢✉❧ ✐♥ ❡st✐♠❛t✐♥❣ r❛♥❞♦♠ Σ

♣r✐♦rs ✺✺ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s

❊♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥✿ ❳ ∼ Em (iT)

❞✐r❡❝t ❡①t❡♥s✐♦♥ ♦❢ ✉♥✐✈❛r✐❛t❡ ❝❛s❡

X ∼ Em

fiT (①) ≡ ✶ T

T

• t=✶

δ(①t) (①) FiT (①) ≡ ✶ T

T

• t=✶

H(①t) (①) φiT (ω) ≡ ✶ T

T

• t=✶

eiω′①t ♠♦♠❡♥ts ✐♥❝❧✉❞❡

✶

s❛♠♣❧❡ ♠❡❛♥✿ ˆ EiT ≡ ✶

T

T

t=✶ ①t

✷

s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡✿ ˆ Cov iT ≡ ✶

T

T

t=✶

• ①t − ˆ

EiT ①t − ˆ EiT ′

✺✻ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❞✐str✐❜✉t✐♦♥s

❊❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s✿ ❳ ∼ El (µ, Σ, gN)

❤✐❣❤❧② s②♠♠❡tr✐❝❛❧✱ ❛♥❛❧②t✐❝❛❧❧② tr❛❝t❛❜❧❡✱ ✢❡①✐❜❧❡ ❳ ✐s ❡❧❧✐♣t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ✇✐t❤ ❧♦❝❛t✐♦♥ ♣❛r❛♠❡t❡r µ ❛♥❞ s❝❛tt❡r ♠❛tr✐① Σ ✐❢ ✐ts ✐s♦✲♣r♦❜❛❜✐❧✐t② ❝♦♥t♦✉rs ❢♦r♠ ❡❧❧✐♣s♦✐❞s ❝❡♥tr❡❞ ❛t µ ✇❤♦s❡ ♣r✐♥❝✐♣❛❧ ❛①❡s✬ ❧❡♥❣t❤s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ sq✉❛r❡ r♦♦ts ♦❢ Σ✬s ❡✐❣❡♥✈❛❧✉❡s ❡❧❧✐♣t✐❝❛❧ ♣❞❢ ♠✉st ❜❡ fµ,Σ (①) = |Σ|− ✶

✷ gN

• Ma✷ (①, µ, Σ)
• ✇❤❡r❡ gN (·) ≥ ✵ ✐s ❛ ❣❡♥❡r❛t♦r ❢✉♥❝t✐♦♥ r♦t❛t❡❞ t♦ ❢♦r♠ t❤❡

❞✐str✐❜✉t✐♦♥✳ ❡①❛♠♣❧❡s ✐♥❝❧✉❞❡✿ ✉♥✐❢♦r♠ ✭s♦♠❡t✐♠❡s✮✱ ♥♦r♠❛❧✱ ❙t✉❞❡♥t t✱ ❈❛✉❝❤② ❛✣♥❡ tr❛♥s❢♦r♠❛t✐♦♥s✿ ❢♦r ❛♥② K✲✈❡❝t♦r ❛✱ K × N ♠❛tr✐① ❇✱ ❛♥❞ t❤❡ r✐❣❤t ❣❡♥❡r❛t♦r gK✱ ❳ ∼ El (µ, Σ, gN) ⇒ ❛ + ❇❳ ∼ El

• ❛ + ❇µ, ❇Σ❇′, gK
• ❝♦rr❡❧❛t✐♦♥ ❝❛♣t✉r❡s ❛❧❧ ❞❡♣❡♥❞❡♥❝❡ str✉❝t✉r❡ ✭❝♦♣✉❧❛ ❛❞❞s ♥♦t❤✐♥❣✮

✺✽ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❞✐str✐❜✉t✐♦♥s

❙t❛❜❧❡ ❞✐str✐❜✉t✐♦♥s

❧❡t ❳, ❨ ❛♥❞ ❩ ❜❡ ■■❉ r✈s❀ t❤❡✐r ❞✐str✐❜✉t✐♦♥ ✐s st❛❜❧❡ ✐❢ ❢♦r ❛♥② ❝♦♥st❛♥ts α, β > ✵ t❤❡r❡ ❡①✐st ❝♦♥st❛♥ts γ ❛♥❞ δ > ✵ s✉❝❤ t❤❛t α❳ + β❨ d = γ + δ❩ ❡①❛♠♣❧❡s✿ ♥♦r♠❛❧✱ ❈❛✉❝❤② ✭❜✉t ♥♦t ❧♦❣♥♦r♠❛❧✱ ♦r ❣❡♥❡r✐❝ ❙t✉❞❡♥t t✮ ❝❧♦s❡❞ ✉♥❞❡r ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s✱ t❤✉s ❛❧❧♦✇s ❡❛s② ♣r♦❥❡❝t✐♦♥ t♦ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥s st❛❜✐❧✐t② ✐♠♣❧✐❡s ❛❞❞✐t✐✈✐t② ✭t❤❡ s✉♠ ♦❢ t✇♦ ■■❉ r✈s ❜❡❧♦♥❣s t♦ t❤❡ s❛♠❡ ❢❛♠✐❧② ♦❢ ❞✐str✐❜✉t✐♦♥s✮✱ ❜✉t ♥♦t t❤❡ r❡✈❡rs❡

❊①❛♠♣❧❡

✶ st❛❜❧❡ ⇒ ❛❞❞✐t✐✈❡✿ X, Y , Z ∼ NID

• ✶, σ✷

⇒ X + Y d = ✷ − √ ✷ + √ ✷Z

✷ ❛❞❞✐t✐✈❡ ⇒ st❛❜❧❡✿

❳, ❨ , ❩ ∼ WID (ν, Σ) ⇒ ❳ + ❨ ∼ W (✷ν, Σ)

d

= γ + δ❩

✺✾ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❞✐str✐❜✉t✐♦♥s

■♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ❞✐str✐❜✉t✐♦♥s

t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ r✈ ❳ ✐s ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ✐❢✱ ❢♦r ❛♥② ✐♥t❡❣❡r T ❳ d = ❨✶ + · · · + ❨T ❢♦r s♦♠❡ ■■❉ r✈s ❨✶, . . . , ❨T ❡①❛♠♣❧❡s ✐♥❝❧✉❞❡✿ ❛❧❧ ❡❧❧✐♣t✐❝❛❧✱ ❣❛♠♠❛✱ ▲♦❣◆ ✭❜✉t ♥♦t ❲✐s❤❛rt ❢♦r N > ✶✮ s❤❛❧❧ s❡❡✿ ❛ss✐sts ✐♥ ♣r♦❥❡❝t✐♦♥ t♦ ❛r❜✐tr❛r② ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥s ✭❡✳❣✳ ❛♥② T✮

✻✵ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❞✐str✐❜✉t✐♦♥s

▲❡❝t✉r❡ ✸ ❡①❡r❝✐s❡s

▼❡✉❝❝✐ ❡①❡r❝✐s❡s

♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✶✳✷✳✻✱ ✶✳✷✳✼✱ ✶✳✸✳✶✱ ✶✳✸✳✹✱ ✷✳✶✳✸ ▼❆❚▲❆❇✿ ✶✳✷✳✽✱ ✶✳✸✳✷✱ ✶✳✸✳✸✱ ✷✳✶✳✶✱ ✷✳✶✳✷

♣r♦❥❡❝t

s♣❡❝✐❢② ❛ ♣♦rt❢♦❧✐♦ ✉s✐♥❣ ②♦✉r t❤r❡❡ ❛ss❡ts✳ ■t ♥❡❡❞ ♥♦t ❜❡ ♦♣t✐♠❛❧✳ ❯♣❞❛t❡ ✐t ❛s t❤❡ t❡r♠ ❣♦❡s ♦♥✳

✻✶ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s

■♥tr♦❞✉❝t✐♦♥

t❤❡ ❝♦♣✉❧❛ ✐s ❛ st❛♥❞❛r❞✐③❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ♣✉r❡❧② ❥♦✐♥t ❢❡❛t✉r❡s ♦❢ ❛ ♠✉❧t✐✈❛r✐❛t❡ ❞✐str✐❜✉t✐♦♥✱ ✇❤✐❝❤ ✐s ♦❜t❛✐♥❡❞ ❜② ✜❧t❡r✐♥❣ ♦✉t ❛❧❧ t❤❡ ♣✉r❡❧② ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❢❡❛t✉r❡s✱ ♥❛♠❡❧② t❤❡ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ❡❛❝❤ ❡♥tr② Xn✳ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ♣✳✹✵✮ ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ❈❤ ✼✮ ❣♦❡s ✐♥t♦ ♠♦r❡ ❞❡t❛✐❧ t❤❛♥ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ❈❤ ✷✮ ♦♥ ❝♦♣✉❧❛s

♠♦r❡ ♠❛t❡r✐❛❧ ❛❜♦✉t t❤❡ ❜♦♦❦ ✐s ❛✈❛✐❧❛❜❧❡ ❛t ✇✇✇✳qr♠t✉t♦r✐❛❧✳♦r❣

s❡❡ ❊♠❜r❡❝❤ts ✭✷✵✵✾✮ ❢♦r t❤♦✉❣❤ts ♦♥ t❤❡ ✏❝♦♣✉❧❛ ❝r❛③❡✑✱ ❢r♦♠ ♦♥❡ ♦❢ ✐ts ♣✐♦♥❡❡rs✱ ❛♥❞ ❛ ✏♠✉st✲r❡❛❞✑ ❢♦r ❝♦♥t❡①t t❤❡ ❝❧❛ss✐❝ t❡①t ✐s ◆❡❧s❡♥ ✭✷✵✵✻✮❀ ✐t ❝♦♥t❛✐♥s ✇♦r❦❡❞ ❡①❛♠♣❧❡s ❛♥❞ s❡t q✉❡st✐♦♥s✱ ❛♥❞ ❤❛s t❤❡ s♣❛❝❡ t♦ ♣r♦♣❡r❧② ❞❡✈❡❧♦♣ t❤❡ ❜❛s✐❝ ❝♦♥❝❡♣ts ❛ ✷✵✵✾ ✇✐r❡❞✳❝♦♠ ❛rt✐❝❧❡ ❜❧❛♠❡❞ t❤❡ ●❛✉ss✐❛♥ ❝♦♣✉❧❛ ❢♦r♠✉❧❛ ❢♦r ✏❦✐❧❧✐♥❣✑ ❲❛❧❧ ❙tr❡❡t

✻✸ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s

❈♦♣✉❧❛s ❞❡✜♥❡❞

❉❡✜♥✐t✐♦♥

❆♥ N✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♣✉❧❛✱ ❯✱ ✐s ❞❡✜♥❡❞ ♦♥ [✵, ✶]N❀ ✐ts ❏❉❋✱ F❯✱ ❤❛s st❛♥❞❛r❞ ✉♥✐❢♦r♠ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s✳

❝♦♣✉❧❛ ❡①❛♠♣❧❡

❊♠❜r❡❝❤ts ✭✷✵✵✾✱ ♣✳✻✹✵✮ ♥♦t❡s t❤❛t ♦t❤❡r st❛♥❞❛r❞✐s❛t✐♦♥s t❤❛♥ t❤❡ ❝♦♣✉❧❛✬s t♦ U ([✵, ✶]) ♠❛② ❜❡ ♠♦r❡ ✉s❡❢✉❧ ✉♥❞❡r ❝❡rt❛✐♥ ❝✐r❝✉♠st❛♥❝❡s

✻✹ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s

❙❦❧❛r✬s t❤❡♦r❡♠

❚❤❡♦r❡♠ ✭❙❦❧❛r✱ ✶✾✺✾✮

▲❡t F❳ ❜❡ ❛ ❏❉❋ ✇✐t❤ ♠❛r❣✐♥❛❧s✱ FX✶, . . . , FXN✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♣✉❧❛✱ ❯✱ ✇✐t❤ ❏❉❋ F❯ : [✵, ✶]N → [✵, ✶] s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ x✶, . . . , xN ∈ R✱ F❳ (①) = F❯ (FX✶ (x✶) , . . . , FXN (xN)) . ✭✶✮ ■❢ t❤❡ ♠❛r❣✐♥❛❧s ❛r❡ ❝♦♥t✐♥✉♦✉s✱ F❯ ✐s ✉♥✐q✉❡✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ ❯ ✐s ❛ ❝♦♣✉❧❛ ❛♥❞ FX✶, . . . , FXN ❛r❡ ✉♥✐✈❛r✐❛t❡ ❈❉❋s✱ t❤❡♥ F❳✱ ❞❡✜♥❡❞ ✐♥ ❡q✉❛t✐♦♥ ✶ ✐s ❛ ❏❉❋ ✇✐t❤ ♠❛r❣✐♥❛❧s FX✶, . . . , FXN✳ ❯s❡❢✉❧ t♦ ❞❡❝♦♠♣♦s❡ r✈ ✐♥t♦ ♠❛r❣✐♥❛❧s ❛♥❞ ❝♦♣✉❧❛✿

✶ ♠❛② ❤❛✈❡ ♠♦r❡ ❝♦♥✜❞❡♥❝❡ ✐♥ ♠❛r❣✐♥❛❧s t❤❛♥ ❏❉❋

❡✳❣✳ ♠✉❧t✐✈❛r✐❛t❡ t ✇✐t❤ ❞✐✛❡r✐♥❣ t❛✐❧✲t❤✐❝❦♥❡ss ♣❛r❛♠❡t❡rs ❝❛♥ ♠♦❞✐❢② ❥♦✐♥t ❞✐str✐❜✉t✐♦♥s ♦❢ ❡①tr❡♠❡ ✈❛❧✉❡s

✷ ❝❛♥ ❡①♣❡r✐♠❡♥t ✇✐t❤ s❤♦❝❦s✿ ✐❞✐♦s②♥❝r❛t✐❝ ✈✐❛ ♠❛r❣✐♥❛❧s✱ ❝♦♠♠♦♥ ✈✐❛

❝♦♣✉❧❛ ▼❡✉❝❝✐ ✭✷✵✵✺✱ ✭✷✳✸✵✮✮ r❡❧❛t❡s f❳ t♦ f❯✿ s♦♠❡t✐♠❡s ♠♦r❡ ✉s❡❢✉❧

✻✺ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s

Pr♦❜❛❜✐❧✐t② ❛♥❞ q✉❛♥t✐❧❡ tr❛♥s❢♦r♠❛t✐♦♥s

■❢ ✇❛♥t t♦ st♦❝❤❛st✐❝❛❧❧② s✐♠✉❧❛t❡ Z✱ ❜✉t X ✐s ❡❛s✐❡r t♦ ❣❡♥❡r❛t❡✱ ❝❛♥ tr❛♥s❢♦r♠ ❛♥② r✈ X ✇✐t❤ ❝♦♥t✐♥✉♦✉s ❈❉❋ ✐♥t♦ ❛♥② ♦t❤❡r Z ✈✐❛ ❛ ♥❡✇ r✈ U

❚❤❡♦r❡♠ ✭Pr♦♣♦s✐t✐♦♥ ✼✳✷ ▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts ✭✷✵✶✺✮❀ ▼❡✉❝❝✐ ✷✳✷✺ ✲ ✷✳✷✼✮

▲❡t FX ❜❡ ❛ ❈❉❋ ❛♥❞ ❧❡t QX ❞❡♥♦t❡ ✐ts ✐♥✈❡rs❡✳ ❚❤❡♥

✶ ✐❢ X ❤❛s ❛ ❝♦♥t✐♥✉♦✉s ✉♥✐✈❛r✐❛t❡ ❈❉❋✱ FX✱ t❤❡♥ FX (X) ∼ U ([✵, ✶]) ♣r♦♦❢ ✷ ✐❢ U ≡ FX (X) d

= FZ (Z) ∼ U ([✵, ✶])✱ t❤❡♥ Z d = QZ (U) t❤❡ ♥❡✇ r✈✱ U ✐s t❤❡ ❣r❛❞❡ ♦❢ X ♥♦✇ ❤❛✈❡ ✸r❞ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r ❝♦♣✉❧❛s✿ ❯✱ t❤❡ ❝♦♣✉❧❛ ♦❢ ❛ ♠✉❧t✐✈❛r✐❛t❡ r✈✱ ❳✱ ✐s t❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ ✐ts ❣r❛❞❡s (U✶, . . . , UN)′ ≡ (FX✶ (X✶) , . . . , FXN (XN))′

✻✻ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s

■♥❞❡♣❡♥❞❡♥❝❡ ❝♦♣✉❧❛

✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ r✈s ⇔ ❏❉❋ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡✐r ✉♥✐✈❛r✐❛t❡ ❈❉❋s ❛♣♣❧②✐♥❣ ❙❦❧❛r✬s t❤❡♦r❡♠ t♦ ✐♥❞❡♣❡♥❞❡♥t r✈s✱ X✶, . . . , XN F❳ (①) =

N

• n=✶

FXn (xn) = F❯ (FX✶ (x✶) , . . . , FXN (xN)) t❤✉s✱ s✉❜st✐t✉t✐♥❣ FXn (xn) = un✱ ♣r♦✈✐❞❡s t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ❝♦♣✉❧❛ Π (✉) ≡ F❯ (u✶, . . . , uN) =

N

• n=✶

un ✇❤✐❝❤ ✐s ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ♦♥ t❤❡ ✉♥✐t ❤②♣❡r✲❝✉❜❡✱ ✇✐t❤ ❛ ❤♦r✐③♦♥t❛❧ ♣❞❢✱ π (✉) = ✶ ❙❝❤✇❡✐③❡r✲❲♦❧❢ ♠❡❛s✉r❡s ♦❢ ❞❡♣❡♥❞❡♥❝❡ ✭✐♥❞❡①❡❞ ❜② p ✐♥ Lp✲♥♦r♠✮✿ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❛ ❝♦♣✉❧❛ ❛♥❞ t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ❝♦♣✉❧❛

✻✼ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s

❙tr✐❝t❧② ✐♥❝r❡❛s✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♠❛r❣✐♥❛❧s

r❡❝❛❧❧✿ ❝♦rr❡❧❛t✐♦♥ ♦♥❧② ✐♥✈❛r✐❛♥t ✉♥❞❡r ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s

❚❤❡♦r❡♠ ✭Pr♦♣♦s✐t✐♦♥ ✼✳✼ ▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts ✭✷✵✶✺✮✮

▲❡t (X✶, . . . , XN) ❜❡ ❛ r✈ ✇✐t❤ ❝♦♥t✐♥✉♦✉s ♠❛r❣✐♥❛❧s ❛♥❞ ❝♦♣✉❧❛ ❯✱ ❛♥❞ ❧❡t g✶, . . . , gN ❜❡ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥s✳ ❚❤❡♥ (g✶ (X✶) , . . . , gN (XN)) ❛❧s♦ ❤❛s ❝♦♣✉❧❛ ❯✳ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤✐s ✐s t❤❡ ❝♦✲♠♦♥♦t♦♥✐❝✐t② ❝♦♣✉❧❛

❧❡t t❤❡ r✈s X✶, . . . , XN ❤❛✈❡ ❝♦♥t✐♥✉♦✉s ❞❢s t❤❛t ❛r❡ ♣❡r❢❡❝t❧② ♣♦s✐t✐✈❡❧② ❞❡♣❡♥❞❡♥t✱ s♦ t❤❛t Xn = gn (X✶) ❛❧♠♦st s✉r❡❧② ❢♦r ❛❧❧ n ∈ {✷, . . . , N} ❢♦r str✐❝t❧② ✐♥❝r❡❛s✐♥❣ gn (·) ❝♦✲♠♦♥♦t♦♥✐❝✐t② ❝♦♣✉❧❛ ✐s t❤❡♥ M (✉) ≡ ♠✐♥ {u✶, . . . , uN} ✇❤❡r❡ t❤❡ ❏❉❋ ♦❢ t❤❡ r✈ (U, . . . , U) ✐s s✳t✳ U ∼ U ([✵, ✶]) ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✷✷✻✮

✻✽ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s

❋ré❝❤❡t✲❍♦❡✛❞✐♥❣ ❜♦✉♥❞s

❝♦✲♠♦♥♦t♦♥✐❝✐t② ❝♦♣✉❧❛✱ M✱ ✐s ❋ré❝❤❡t✲❍♦❡✛❞✐♥❣ ✉♣♣❡r ❜♦✉♥❞ ❋ré❝❤❡t✲❍♦❡✛❞✐♥❣ ❧♦✇❡r ❜♦✉♥❞✱ W ✱ ✐s♥✬t ❝♦♣✉❧❛ ❢♦r N > ✷✿ W (✉) ≡ ♠❛①

• ✶ − N +

N

• n=✶

un, ✵

• ❛♥② ❝♦♣✉❧❛✬s ❈❉❋ ✜ts ❜❡t✇❡❡♥

t❤❡s❡ W (✉) ≤ F❯ (✉) ≤ M (✉) ✇❤✐❝❤ ❝♦♣✉❧❛ ✐s ✷♥❞ ✜❣✉r❡❄ ❘ ❝♦❞❡✿ ❍är❞❧❡ ❛♥❞ ❖❦❤r✐♥ ✭✷✵✶✵✮

✻✾ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s

❆ ❝❛❧❧ ♦♣t✐♦♥

❊①❛♠♣❧❡

❈♦♥s✐❞❡r t✇♦ st♦❝❦ ♣r✐❝❡s✱ t❤❡ r✈s ❳ = (X✶, X✷)✱ ❛♥❞ ❛ ❊✉r♦♣❡❛♥ ❝❛❧❧ ♦♣t✐♦♥ ♦♥ t❤❡ ✜rst ✇✐t❤ str✐❦❡ ♣r✐❝❡ K✳ ❚❤❡ ♣❛②♦✛ ♦♥ t❤✐s ♦♣t✐♦♥ ✐s t❤❡r❡❢♦r❡ ❛❧s♦ ❛ r✈✱ C✶ ≡ ♠❛① {X✶ − K, ✵}✳ ❚❤✉s✱ C✶ ❛♥❞ X✶ ❛r❡ ❝♦✲♠♦♥♦t♦♥✐❝❀ t❤❡✐r ❝♦♣✉❧❛ ✐s M✱ t❤❡ ❝♦✲♠♦♥♦t♦♥✐❝✐t② ❝♦♣✉❧❛✳ ❋✉rt❤❡r✱ (X✶, X✷) ❛♥❞ (C✶, X✷) ❛r❡ ❛❧s♦ ❝♦✲♠♦♥♦t♦♥✐❝❀ t❤❡ ❝♦♣✉❧❛ ♦❢ (X✶, X✷) ✐s t❤❡ s❛♠❡ ❛s t❤❛t ♦❢ (C✶, X✷)✳ ❲❤❛t t❡❝❤♥✐❝❛❧ ❞❡t❛✐❧ ✐s t❤❡ ❛❜♦✈❡ ♠✐ss✐♥❣❄ ❍♦✇ ✐s t❤✐s ♦✈❡r❝♦♠❡❄

❝♦✲♠♦♥♦t♦♥✐❝ ❛❞❞✐t✐✈✐t② ✼✵ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t

❈♦♥❝❡♣t✉❛❧ ♦✈❡r✈✐❡✇

▼❡✉❝❝✐ ✭✷✵✵✺✮ ✐❞❡♥t✐✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ st❡♣s ❢♦r ❜✉✐❧❞✐♥❣ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ❤✐st♦r✐❝❛❧ ♣❡r❢♦r♠❛♥❝❡ ❛♥❞ ❢✉t✉r❡ ❞✐str✐❜✉t✐♦♥s

✶ ❞❡t❡❝t✐♥❣ t❤❡ ✐♥✈❛r✐❛♥ts

✇❤❛t ♠❛r❦❡t ✈❛r✐❛❜❧❡s ❝❛♥ ❜❡ ♠♦❞❡❧❧❡❞ ❛s ■■❉ r✈s❄

✷ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♥✈❛r✐❛♥ts

❤♦✇ ❢r❡q✉❡♥t❧② ❞♦ t❤❡s❡ ❝❤❛♥❣❡ ✭q✳✈✳ ❇❛✉❡r ❛♥❞ ❇r❛✉♥ ✭✷✵✶✵✮✮

✸ ♣r♦❥❡❝t✐♥❣ t❤❡ ✐♥✈❛r✐❛♥ts ✐♥t♦ t❤❡ ❢✉t✉r❡ ✹ ♠❛♣♣✐♥❣ t❤❡ ✐♥✈❛r✐❛♥ts ✐♥t♦ t❤❡ ♠❛r❦❡t ♣r✐❝❡s

❆s t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ ❵♠♦st✬ r❛♥❞♦♠♥❡ss ♠❛② ❜❡ ♠✉❝❤ ❧❡ss t❤❛♥ t❤❛t ♦❢ t❤❡ ♣♦rt❢♦❧✐♦ s♣❛❝❡✱ ❞✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ t❡❝❤♥✐q✉❡s ✇✐❧❧ ❡♥❤❛♥❝❡ tr❛❝t❛❜✐❧✐t②

✼✶ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❙t②❧✐s❡❞ ❢❛❝ts

❯♥✐✈❛r✐❛t❡ st②❧✐s❡❞ ❢❛❝ts

• ✐✈❡♥ ❛♥ ❛ss❡t ♣r✐❝❡ Pt✱ ❧❡t ✐ts ❝♦♠♣♦✉♥❞ r❡t✉r♥ ❛t t✐♠❡ t ❢♦r ❤♦r✐③♦♥ τ ❜❡

Ct,τ ≡ ❧♥ Pt Pt−τ ❚❤❡♥✱ ❢♦❧❧♦✇✐♥❣ ▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts ✭✷✵✶✺✱ ➓✸✳✶✮✿

✶ s❡r✐❡s ♦❢ ❝♦♠♣♦✉♥❞ r❡t✉r♥s ❛r❡ ♥♦t ■■❉✱ ❜✉t s❤♦✇ ❧✐tt❧❡ s❡r✐❛❧

❝♦rr❡❧❛t✐♦♥ ❛❝r♦ss ❞✐✛❡r❡♥t ❧❛❣s

✐❢ ♥♦t ■■❉✱ t❤❡♥ ♣r✐❝❡s ❞♦♥✬t ❢♦❧❧♦✇ r❛♥❞♦♠ ✇❛❧❦ ✐❢ ♥❡✐t❤❡r ■■❉ ♥♦r ♥♦r♠❛❧✱ ❇❧❛❝❦✲❙❝❤♦❧❡s✲▼❡rt♦♥ ♣r✐❝✐♥❣ ✐s ✐♥ tr♦✉❜❧❡

✷ ✈♦❧❛t✐❧✐t② ❝❧✉st❡r✐♥❣✿ s❡r✐❡s ♦❢ |Ct,τ| ♦r C ✷

t,τ s❤♦✇ ♣r♦❢♦✉♥❞ s❡r✐❛❧

❝♦rr❡❧❛t✐♦♥

✸ ❝♦♥❞✐t✐♦♥❛❧ ✭♦♥ ❛♥② ❤✐st♦r②✮ ❡①♣❡❝t❡❞ r❡t✉r♥s ❛r❡ ❝❧♦s❡ t♦ ③❡r♦ ✹ ✈♦❧❛t✐❧✐t② ❛♣♣❡❛rs t♦ ✈❛r② ♦✈❡r t✐♠❡ ✺ ❡①tr❡♠❡ r❡t✉r♥s ❛♣♣❡❛r ✐♥ ❝❧✉st❡rs ✻ r❡t✉r♥s s❡r✐❡s ❛r❡ ❧❡♣t♦❦✉rt✐❝ ✭❤❡❛✈②✲t❛✐❧❡❞✮

❛s ❤♦r✐③♦♥ ✐♥❝r❡❛s❡s✱ r❡t✉r♥s ♠♦r❡ ■■❉✱ ❧❡ss ❤❡❛✈②✲t❛✐❧❡❞

✼✸ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❙t②❧✐s❡❞ ❢❛❝ts

▼✉❧t✐✈❛r✐❛t❡ st②❧✐s❡❞ ❢❛❝ts

• ✐✈❡♥ ❛ ✈❡❝t♦r ♦❢ ❛ss❡t ♣r✐❝❡s Pt✱ ❧❡t ✐ts ❝♦♠♣♦✉♥❞ r❡t✉r♥ ❛t t✐♠❡ t ❢♦r

❤♦r✐③♦♥ τ ❜❡ ❞❡✜♥❡❞ ❝♦♠♣♦♥❡♥t✲✇✐s❡ ❛s ❈t,τ ≡ ❧♥ Pt Pt−τ ❋♦❧❧♦✇✐♥❣ ▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts ✭✷✵✶✺✱ ➓✸✳✷✮

✶ ❈t,τ s❡r✐❡s s❤♦✇ ❧✐tt❧❡ ❡✈✐❞❡♥❝❡ ♦❢ ✭s❡r✐❛❧✮ ❝r♦ss✲❝♦rr❡❧❛t✐♦♥✱ ❡①❝❡♣t ❢♦r

❝♦♥t❡♠♣♦r❛♥❡♦✉s r❡t✉r♥s

✷ |❈t,τ| s❡r✐❡s s❤♦✇ ♣r♦❢♦✉♥❞ ❡✈✐❞❡♥❝❡ ♦❢ ✭s❡r✐❛❧✮ ❝r♦ss✲❝♦rr❡❧❛t✐♦♥ ✸ ❝♦rr❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❝♦♥t❡♠♣♦r❛♥❡♦✉s r❡t✉r♥s ✈❛r② ♦✈❡r t✐♠❡ ✹ ❡①tr❡♠❡ r❡t✉r♥s ✐♥ ♦♥❡ s❡r✐❡s ♦❢t❡♥ ❝♦✐♥❝✐❞❡ ✇✐t❤ ❡①tr❡♠❡ r❡t✉r♥s ✐♥

s❡✈❡r❛❧ ♦t❤❡r s❡r✐❡s

✼✹ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

▼❛r❦❡t ✐♥✈❛r✐❛♥ts

♠❛r❦❡t ✐♥✈❛r✐❛♥ts✱ Xt✱ ❛r❡

✶

■■❉ r✈s

✷

t❛❦✐♥❣ ♦♥ r❡❛❧✐s❡❞ ✈❛❧✉❡s xt ❛t t✐♠❡ t

t❤❡② ❛r❡ t✐♠❡ ❤♦♠♦❣❡♥❡♦✉s ✐❢ t❤❡ ■■❉ ❞✐str✐❜✉t✐♦♥ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❛ r❡❢❡r❡♥❝❡ ❞❛t❡✱ ˜ t ✐♥✈❛r✐❛♥ts ❧✐❦❡ t❤✐s ♠❛❦❡ ✐t ❵❡❛s②✬ t♦ ❢♦r❡❝❛st ❤♦✇ t❡st ❢♦r ■■❉ ✭❈❛♠♣❜❡❧❧✱ ▲♦✱ ❛♥❞ ▼❛❝❑✐♥❧❛②✱ ✶✾✾✼✱ ❈❤❛♣t❡r ✷✮❄

✐♥ ♣❛rt✐❝✉❧❛r✱ ❤♦✇ ♣♦s✐t t❤❡ r✐❣❤t H✶❄ t❡sts ❛❣❛✐♥st ♣❛rt✐❝✉❧❛r H✶✬s ♦❢t❡♥ ♠✐ss❡❞ ♥♦♥✲❧✐♥❡❛r ❞❡t❡r♠✐♥✐st✐❝ r❡❧❛t✐♦♥s❤✐♣s ❡✳❣✳ ❧♦❣✐st✐❝ ♠❛♣✱ xt+✶ = rxt (✶ − xt) ❛♥❞ t❡♥t ♠❛♣✱ xt+✶ =

• µxt

✐❢ xt < ✶

✷

µ (✶ − xt) ♦t❤❡r✇✐s❡

• ❇❉❙✭▲✮ t❡st ✭❇r♦❝❦ ❡t ❛❧✳✱ ✶✾✾✻✮ ❞❡s✐❣♥❡❞ t♦ ❝❛♣t✉r❡ t❤✐s✱ ❜✉t ❢❛✐❧s ✐♥

t❤❡ ♣r❡s❡♥❝❡ ♦❢ r❡❛❧ ♥♦✐s❡❀ ♥♦t ♦❢t❡♥ ✉s❡❞ ❞✉❡ t♦ str♦♥❣ t❤❡♦r❡t✐❝❛❧ ♣r✐♦rs ♦♥ H✶

✇❡ t❤❡r❡❢♦r❡ ♣r❡s❡♥t t✇♦ ❤❡✉r✐st✐❝ t❡sts ✭q✳✈✳ ▼❡✉❝❝✐✱ ✷✵✵✾✱ ➓✷✮

✼✻ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❍❡✉r✐st✐❝ t❡st ✶✿ ❝♦♠♣❛r❡ s♣❧✐t s❛♠♣❧❡ ❤✐st♦❣r❛♠s

❜② t❤❡ ●❧✐✈❡♥❦♦✲❈❛♥t❡❧❧✐ t❤❡♦r❡♠✱ ❡♠♣✐r✐❝❛❧ ♣❞❢ → tr✉❡ ♣❞❢ ❛s t❤❡ ♥✉♠❜❡r ♦❢ ■■❉ ♦❜s❡r✈❛t✐♦♥s ❣r♦✇s s♣❧✐t t❤❡ t✐♠❡ s❡r✐❡s ✐♥ ❤❛❧❢ ❛♥❞ ❝♦♠♣❛r❡ t❤❡ t✇♦ ❤✐st♦❣r❛♠s ✇❤❛t s❤♦✉❧❞ t❤❡ t✇♦ ❤✐st♦❣r❛♠s ❧♦♦❦ ❧✐❦❡ ✐❢ ■■❉❄

✼✼ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❉♦ st♦❝❦ ♣r✐❝❡s✱ Pt✱ ♣❛ss t❤❡ ❤✐st♦❣r❛♠ t❡st❄

❈❛✈❡❛t✿ ❛♣♣❛r❡♥t s✐♠✐❧❛r✐t② ❝❤❛♥❣❡s ✇✐t❤ ❜✐♥ s✐③❡ ❝❤♦✐❝❡ ❆❧❧ ❞❛t❛✿ ❚❍❆❘●❊❙✿■❉ ✵✶✴✵✶✴✵✼ ✕ ✶✵✴✵✾✴✵✾

✼✽ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❉♦ ❧✐♥❡❛r st♦❝❦ r❡t✉r♥s✱ Lt,τ✱ ♣❛ss t❤❡ ❤✐st♦❣r❛♠ t❡st❄

▲✐♥❡❛r r❡t✉r♥s ❛r❡ Lt,τ ≡

Pt Pt−τ − ✶

✼✾ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❉♦ ❝♦♠♣♦✉♥❞ st♦❝❦ r❡t✉r♥s✱ Ct,τ✱ ♣❛ss t❤❡ ❤✐st♦❣r❛♠ t❡st❄

❈♦♠♣♦✉♥❞ r❡t✉r♥s ❛r❡ Ct,τ ≡ ❧♥

Pt Pt−τ

✽✵ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❍❡✉r✐st✐❝ t❡st ✷✿ ♣❧♦t xt ✈ xt−˜

τ

♣❧♦t xt ✈ xt−˜

τ✱ ✇❤❡r❡ ˜

τ ✐s t❤❡ ❡st✐♠❛t✐♦♥ ✐♥t❡r✈❛❧ ✇❤❛t s❤♦✉❧❞ t❤❡ ♣❧♦t ❧♦♦❦ ❧✐❦❡ ✐❢ ■■❉❄

s②♠♠❡tr✐❝ ❛❜♦✉t t❤❡ ❞✐❛❣♦♥❛❧✿ ✐❢ ■■❉✱ ❞♦❡s♥✬t ♠❛tt❡r ✐❢ ♣❧♦t xt ✈ xt−˜

τ

♦r xt−˜

τ ✈ xt

❝✐r❝✉❧❛r✿ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❡❧❧✐♣s♦✐❞ ✇✐t❤ ♠❡❛♥ (µ, µ)✱ ✈❛r✐❛♥❝❡ s❛♠❡ ✐♥ ❡❛❝❤ ❞✐r❡❝t✐♦♥✱ ❛❧✐❣♥❡❞ ✇✐t❤ ❝♦♦r❞✐♥❛t❡ ❛①❡s ❛s ❝♦✈❛r✐❛♥❝❡ ③❡r♦ ✭❞✉❡ t♦ ✐♥❞❡♣❡♥❞❡♥❝❡✮ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ♣✳✺✺✮

❤✐♥t ✽✶ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❉♦ st♦❝❦ ♣r✐❝❡s✱ Pt✱ ♣❛ss t❤❡ ❧❛❣❣❡❞ ♣❧♦t t❡st❄

❲❤❛t ❞♦❡s t❤✐s t❡❧❧ ✉s ❛❜♦✉t st♦❝❦ ♣r✐❝❡s❄

✽✷ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❉♦ ❧✐♥❡❛r st♦❝❦ r❡t✉r♥s✱ Lt,τ✱ ♣❛ss t❤❡ ❧❛❣❣❡❞ ♣❧♦t t❡st❄

❲❤❛t ❞♦ ✇❡ ❡①♣❡❝t ❝♦♠♣♦✉♥❞ r❡t✉r♥s t♦ ❧♦♦❦ ❧✐❦❡✱ ❛s ❛ r❡s✉❧t❄

✐♥❞❡♣❡♥❞❡♥❝❡ ✽✸ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❉♦ ❝♦♠♣♦✉♥❞ st♦❝❦ r❡t✉r♥s✱ Ct,τ✱ ♣❛ss t❤❡ ❧❛❣❣❡❞ ♣❧♦t t❡st❄

❲❤❛t ❞♦ ✇❡ ❡①♣❡❝t t♦t❛❧ r❡t✉r♥s✱ Ht,τ ≡

Pt Pt−τ t♦ ❧♦♦❦

❧✐❦❡❄

✽✹ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

■♥✈❛r✐❛♥ts ❢♦r ❡q✉✐t✐❡s✱ ❝♦♠♠♦❞✐t✐❡s ❛♥❞ ❡①❝❤❛♥❣❡ r❛t❡s

❤❛✈❡ s❡❡♥ ❧✐♥❡❛r✱ ❝♦♠♣♦✉♥❞✱ t♦t❛❧ r❡t✉r♥s ❢♦r ❚❍❆❘●❊❙ ❡q✉✐t② ❢✉♥❞ ♣❛ss t❤❡ ❤❡✉r✐st✐❝ t❡sts ♣r❡❢❡r t♦ ✉s❡ ❝♦♠♣♦✉♥❞ r❡t✉r♥s ❛s

✶

s❤❛❧❧ s❡❡ t❤❛t ❝❛♥ ♠♦r❡ ❡❛s✐❧② ♣r♦❥❡❝t ❞✐str✐❜✉t✐♦♥s t♦ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥

✷

❣r❡❛t❡r s②♠♠❡tr② ❢❛❝✐❧✐t❛t❡s ♠♦❞❡❧❧✐♥❣ ❜② ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s

∆ ❨❚▼

✐♥❞✐✈✐❞✉❛❧ ❡q✉✐t✐❡s✱ ❝♦♠♠♦❞✐t✐❡s✱ ❡①❝❤❛♥❣❡ r❛t❡s ❤❛✈❡ s✐♠✐❧❛r ♣r♦♣❡rt✐❡s✿ ♥♦ t✐♠❡ ❤♦r✐③♦♥s ❦❡② ❛ss✉♠♣t✐♦♥s

✶

❡q✉✐t✐❡s✿ ❡✐t❤❡r ♥♦ ❞✐✈✐❞❡♥❞s✱ ♦r ❞✐✈✐❞❡♥❞s ♣❧♦✉❣❤❡❞ ❜❛❝❦ ✐♥

✷

❣❡♥❡r❛❧❧②✱ ♥♦♥✲♦✈❡r❧❛♣♣✐♥❣ ✲ s❡❡ Wt ✐♥ ▼❡✉❝❝✐✬s ♦♥❧✐♥❡ ❡①❡r❝✐s❡ ✸✳✷✳✶ ✭❖❝t ✷✵✵✾✮ ❛s ❛ ❝♦✉♥t❡r✲❡①❛♠♣❧❡

❛❝❝❡♣t ❝♦♠♣♦✉♥❞ r❡t✉r♥s ❛s ■■❉ ❛s ❡①♣♦s✐t✐♦♥❛❧ ❞❡✈✐❝❡ ✭r❡❝❛❧❧ st②❧✐s❡❞ ❢❛❝ts✮❀ s❡❡ ▼❡✉❝❝✐ ✭✷✵✵✾✮ ❢♦r ♠♦r❡ ❞✐s❝✉ss✐♦♥

✽✺ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

▲❡❝t✉r❡ ✹ ❡①❡r❝✐s❡s

◆❡❧s❡♥ ✭✷✵✵✻✱ ❊①❡r❝✐s❡ ✷✳✶✷✮

▲❡t X ❛♥❞ Y ❜❡ r✈s ✇✐t❤ ❏❉❋ H (x, y) =

• ✶ + e−x + e−y−✶

❢♦r ❛❧❧ x, y ∈ ¯ R✱ t❤❡ ❡①t❡♥❞❡❞ r❡❛❧s✳

✶

s❤♦✇ t❤❛t X ❛♥❞ Y ❤❛✈❡ st❛♥❞❛r❞ ✭✉♥✐✈❛r✐❛t❡✮ ❧♦❣✐st✐❝ ❞✐str✐❜✉t✐♦♥s F (x) =

• ✶ + e−x−✶ ❛♥❞ G (y) =
• ✶ + e−y−✶ .

✷

s❤♦✇ t❤❛t t❤❡ ❝♦♣✉❧❛ ♦❢ X ❛♥❞ Y ✐s C (u, v) =

uv u+v−uv ✳

▼❡✉❝❝✐ ❡①❡r❝✐s❡s

♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✸✳✷✳✶❀ ✐♥ q✉❡st✐♦♥ ✼✳✶✳✶✱ ✇❤② ✐s ❡q✉❛t✐♦♥ ✹✹✵ ♥♦t ❛ t②♣♦❄❀ ✐♥ q✉❡st✐♦♥ ✼✳✸✳✷✱ ❤♦✇ ❞♦ ✇❡ ✏♥♦t✐❝❡ t❤❛t ♥♦r♠❛❧ ♠❛r❣✐♥❛❧s ❬❜♦✉♥❞ t♦❣❡t❤❡r ❜②❪ ❛ ♥♦r♠❛❧ ❝♦♣✉❧❛ ❣✐✈❡ r✐s❡ t♦ ❛ ♥♦r♠❛❧ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥✑❄ ▼❆❚▲❆❇✿ ✷✳✶✳✹✱ ✷✳✷✳✶✱ ✷✳✷✳✷✱ ✷✳✷✳✸✱ ✷✳✷✳✹✱ ✷✳✷✳✺✱ ✷✳✷✳✻✱ ✷✳✷✳✼✱ ✸✳✶✳✶✱ ✸✳✶✳✷✱ ✸✳✶✳✸

♣r♦❥❡❝t

❞♦ ②♦✉r ❛ss❡ts ♣♦ss❡ss t❤❡ r❡❧❡✈❛♥t ✐♥✈❛r✐❛♥t ♣r♦♣❡rt✐❡s❄

✽✻ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❋✐①❡❞ ✐♥❝♦♠❡✿ ③❡r♦✲❝♦✉♣♦♥ ❜♦♥❞s

♠❛❦❡ ♥♦ t❡r♠❧② ♣❛②♠❡♥ts ❛s s✐♠♣❧❡st ❢♦r♠ ♦❢ ❜♦♥❞✱ ❢♦r♠ ❜❛s✐s ❢♦r ❛♥❛❧②s✐s ♦❢ ❜♦♥❞s ✜①❡❞ ✐♥❝♦♠❡ ❛s ❝❡rt❛✐♥ ❬❄❪ ♣❛②♦✉t ❛t ❢❛❝❡ ♦r r❡❞❡♠♣t✐♦♥ ✈❛❧✉❡

✭s❡❡ ❇r✐❣♦✱ ▼♦r✐♥✐✱ ❛♥❞ P❛❧❧❛✈✐❝✐♥✐ ✭✷✵✶✸✮ ❢♦r r✐❝❤❡r r✐s❦ ♠♦❞❡❧❧✐♥❣✮

❜♦♥❞ ♣r✐❝❡ t❤❡♥ Z (E)

t

✱ ✇❤❡r❡ t ≤ E ✐s ❞❛t❡✱ ❛♥❞ E ✐s ♠❛t✉r✐t② ❞❛t❡ ♥♦r♠❛❧✐s❡ Z (E)

E

= ✶ ❛r❡ ❜♦♥❞ ♣r✐❝❡s ✐♥✈❛r✐❛♥ts❄

✶

st♦❝❦ ♣r✐❝❡s ✇❡r❡♥✬t

✷

t✐♠❡ ❤♦♠♦❣❡♥❡✐t② ✈✐♦❧❛t❡❞

❛r❡ r❡t✉r♥s ✭t♦t❛❧✱ s✐♠♣❧❡✱ ❝♦♠♣♦✉♥❞✮ ✐♥✈❛r✐❛♥ts❄

✽✼ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❋✐①❡❞ ✐♥❝♦♠❡✿ ❛ t✐♠❡ ❤♦♠♦❣❡♥❡♦✉s ❢r❛♠❡✇♦r❦

❝♦♥str✉❝t ❛ s②♥t❤❡t✐❝ s❡r✐❡s ♦❢ ❜♦♥❞ ♣r✐❝❡s ✇✐t❤ t❤❡ s❛♠❡ t✐♠❡ t♦ ♠❛t✉r✐t②✱ v✿

✶

Z (E)

t

✭❡✳❣✳ ◆♦✈ ✷✵✶✻ ♣r✐❝❡ ♦❢ ❛ ❜♦♥❞ t❤❛t ♠❛t✉r❡s ✐♥ ❋❡❜ ✷✵✷✶✮

✷

Z (E−˜

τ) t−˜ τ

✭❡✳❣✳ ◆♦✈ ✷✵✶✺ ♣r✐❝❡ ♦❢ ❛ ❜♦♥❞ t❤❛t ♠❛t✉r❡s ✐♥ ❋❡❜ ✷✵✷✵✮

✸

Z (E−✷˜

τ) t−✷˜ τ

✭❡✳❣✳ ◆♦✈ ✷✵✶✹ ♣r✐❝❡ ♦❢ ❛ ❜♦♥❞ t❤❛t ♠❛t✉r❡s ✐♥ ❋❡❜ ✷✵✶✾✮

✹

✳ ✳ ✳

t❛r❣❡t ❞✉r❛t✐♦♥ ❢✉♥❞s✿ ❛♥ ❡st❛❜❧✐s❤❡❞ ✜①❡❞ ✐♥❝♦♠❡ str❛t❡❣② ✭▲❛♥❣❡t✐❡❣✱ ▲❡✐❜♦✇✐t③✱ ❛♥❞ ❑♦❣❡❧♠❛♥✱ ✶✾✾✵✮ ❝❛♥ ♥♦✇ ❞❡✜♥❡ ♣s❡✉❞♦✲r❡t✉r♥s✱ ♦r r♦❧❧✐♥❣ ✭t♦t❛❧✮ r❡t✉r♥s t♦ ♠❛t✉r✐t② R(v)

t,˜ τ ≡

Z (t+v)

t

Z (t−˜

τ+v) t−˜ τ

✇❤❡r❡ ˜ τ ✐s t❤❡ ❡st✐♠❛t✐♦♥ ✐♥t❡r✈❛❧ ✭❡✳❣✳ ❛ ②❡❛r✮ t❤❡s❡ ♣❛ss t❤❡ t✇♦ ❤❡✉r✐st✐❝ t❡sts ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ❋✐❣✉r❡ ✸✳✺✮

✽✽ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❋✐①❡❞ ✐♥❝♦♠❡✿ ②✐❡❧❞ t♦ ♠❛t✉r✐t②

✇❤❛t ✐s t❤❡ ♠♦st ❝♦♥✈❡♥✐❡♥t ✜①❡❞ ✐♥❝♦♠❡ ✐♥✈❛r✐❛♥t t♦ ✇♦r❦ ✇✐t❤❄ ❞❡✜♥❡ Y (v)

t

≡ − ✶

v ❧♥ Z (t+v) t

❛♥❞ ♠❛♥✐♣✉❧❛t❡ t♦ ♦❜t❛✐♥ ❛ ❝♦♠♣♦✉♥❞ r❡t✉r♥✿ vY (v)

t

= − ❧♥ Z (t+v)

t

= ❧♥ ✶ − ❧♥ Z (t+v)

t

= ❧♥ ✶ Z (t+v)

t

= ❧♥ Z (t+v)

t+v

Z (t+v)

t

Y (v)

t

✐s ②✐❡❧❞ t♦ ♠❛t✉r✐t② v❀ ②✐❡❧❞ ❝✉r✈❡ ❣r❛♣❤s Y (v)

t

❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ v ✐❢ ˜ τ ✐s ❛ ②❡❛r ✭st❛♥❞❛r❞✮✱ t❤❡♥ ❨❚▼ ✐s ❧✐❦❡ ❛♥ ❛♥♥✉❛❧✐s❡❞ ②✐❡❧❞ ❝❤❛♥❣❡s ✐♥ ②✐❡❧❞ t♦ ♠❛t✉r✐t② ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ r♦❧❧✐♥❣ r❡t✉r♥s t♦ ♠❛t✉r✐t②✱ X (v)

t,˜ τ ≡ Y (v) t

− Y (v)

t−˜ τ = −✶

v ❧♥ Z (t+v)

t

Z (t−˜

τ+v) t−˜ τ

= −✶ v ❧♥ R(v)

t,˜ τ

✉s✉❛❧❧② ♣❛ss t❤❡ ❤❡✉r✐st✐❝s✱ ❤❛✈❡ s✐♠✐❧❛r❧② ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt✐❡s t♦ ❝♦♠♣♦✉♥❞ r❡t✉r♥s ❢♦r ❡q✉✐t✐❡s

❝♦♠♣♦✉♥❞ r❡t✉r♥s ✽✾ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❉❡r✐✈❛t✐✈❡s

❞❡r✐✈❡❞ ❢r♦♠ ✉♥❞❡r❧②✐♥❣ r❛✇ s❡❝✉r✐t✐❡s ✭❡✳❣✳ st♦❝❦s✱ ③❡r♦✲❝♦✉♣♦♥ ❜♦♥❞s✱ ✳ ✳ ✳ ✮ ✈❛♥✐❧❧❛ ❊✉r♦♣❡❛♥ ♦♣t✐♦♥s ❛r❡ t❤❡ ♠♦st ❧✐q✉✐❞ ❞❡r✐✈❛t✐✈❡s ✭✇❤②❄✮

t❤❡ r✐❣❤t✱ ❜✉t ♥♦t t❤❡ ♦❜❧✐❣❛t✐♦♥✱ t♦ ❜✉② ♦r s❡❧❧ ✳ ✳ ✳ ♦♥ ❡①♣✐r② ❞❛t❡ E ✳ ✳ ✳ ❛♥ ✉♥❞❡r❧②✐♥❣ s❡❝✉r✐t② tr❛❞✐♥❣ ❛t ♣r✐❝❡ Ut ❛t t✐♠❡ t ✳ ✳ ✳ ❢♦r str✐❦❡ ♣r✐❝❡ K

❊①❛♠♣❧❡ ✭❊✉r♦♣❡❛♥ ❝❛❧❧ ♦♣t✐♦♥✮

❚❤❡ ♣r✐❝❡ ♦❢ ❛ ❊✉r♦♣❡❛♥ ❝❛❧❧ ♦♣t✐♦♥ ❛t t✐♠❡ t ≤ E ✐s ♦❢t❡♥ ❡①♣r❡ss❡❞ ❛s C (K,E)

t

≡ C BSM E − t, K, Ut, Z (E)

t

, σ(K,E)

t

• s✳t✳C (K,E)

E

= ♠❛① {UE − K, ✵} ✇❤❡r❡ E − t ✐s t❤❡ t✐♠❡ r❡♠❛✐♥✐♥❣✱ ❛♥❞ σ(K,E)

t

✐s t❤❡ ✈♦❧❛t✐❧✐t② ♦❢ Ut✳ ❚❤❡ ♦♣t✐♦♥ ✐s ✐♥ t❤❡ ♠♦♥❡② ✇❤❡♥ Ut > K✱ ❛t t❤❡ ♠♦♥❡② ✇❤❡♥ Ut = K ❛♥❞ ♦✉t ♦❢ t❤❡ ♠♦♥❡② ♦t❤❡r✇✐s❡✳

✾✵ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❉❡r✐✈❛t✐✈❡s✿ ✈♦❧❛t✐❧✐t②

♣r✐❝✐♥❣ ♦♣t✐♦♥s r❡q✉✐r❡s ❛ ♠❡❛s✉r❡ ♦❢ ✈♦❧❛t✐❧✐t②

✶

❤✐st♦r✐❝❛❧ ♦r r❡❛❧✐s❡❞ ✈♦❧❛t✐❧✐t②✿ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ ❤✐st♦r✐❝❛❧ ✈❛❧✉❡s ♦❢ Ut ✭❡s♣✳ ❆❘❈❍ ♠♦❞❡❧s✮❀ ❜❛❝❦✇❛r❞ ❧♦♦❦✐♥❣ ❜✉t ♠♦❞❡❧✲❢r❡❡

✷

✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t②✿ ❛s t❤❡ ❝❛❧❧ ♦♣t✐♦♥✬s ♣r✐❝❡ ✐♥❝r❡❛s❡s ✐♥ σt✱ t❤❡ ❇❙▼ ♣r✐❝✐♥❣ ❢♦r♠✉❧❛ ❤❛s ❛♥ ✐♥✈❡rs❡✱ ❛❧❧♦✇✐♥❣ ✈♦❧❛t✐❧✐t② t♦ ❜❡ ✐♠♣❧✐❡❞ ❢r♦♠ ♦♣t✐♦♥ ♣r✐❝❡s❀ ❢♦r✇❛r❞ ❧♦♦❦✐♥❣✱ ❜✉t ♠♦❞❡❧✲❞❡♣❡♥❞❡♥t❀ ❡✳❣✳ ❱❳❖

✸

♠♦❞❡❧✲❢r❡❡ ✈♦❧❛t✐❧✐t② ❡①♣❡❝t❛t✐♦♥s✿ r✐s❦✲♥❡✉tr❛❧ ❡①♣❡❝t❛t✐♦♥ ♦❢ ❖❚▼ ♦♣t✐♦♥ ♣r✐❝❡s❀ ❢♦r✇❛r❞ ❧♦♦❦✐♥❣✱ ❧❡ss ♠♦❞❡❧✲❞❡♣❡♥❞❡♥t ✭❜✉t ❛ss✉♠❡s st♦❝❤❛st✐❝ ♣r♦❝❡ss ❞♦❡s♥✬t ❥✉♠♣✮❀ ❡✳❣✳ ❱■❳

❚❛②❧♦r✱ ❨❛❞❛✈✱ ❛♥❞ ❩❤❛♥❣ ✭✷✵✶✵✮ ❝♦♠♣❛r❡ t❤❡ t❤r❡❡ ✈♦❧❛t✐❧✐t② ♠❡❛s✉r❡s ❊✉r♦♣❡✿ ❱❙❚❖❳❳✱ ❱❋❚❙❊✱ ❱❉❆❳✱ ❱❉❆❳✲◆❊❲✱ ❱❈❆❈✱ ❱❙▼■✱ ❱❆❊❳✱ ❱❇❊▲ ❛t✲t❤❡✲♠♦♥❡②✲❢♦r✇❛r❞ ✭❆❚▼❋✮ ✐♠♣❧✐❡❞ ♣❡r❝❡♥t❛❣❡ ✈♦❧❛t✐❧✐t② ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣✿ ✏✐♠♣❧✐❡❞ ♣❡r❝❡♥t❛❣❡ ✈♦❧❛t✐❧✐t② ♦❢ ❛♥ ♦♣t✐♦♥ ✇❤♦s❡ str✐❦❡ ✐s ❡q✉❛❧ t♦ t❤❡ ❢♦r✇❛r❞ ♣r✐❝❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ❛t ❡①♣✐r②✑ ✭▼❡✉❝❝✐✱ ✷✵✵✺✮

❜② ♥♦✲❛r❜✐tr❛❣❡ ❢♦r✇❛r❞ ♣r✐❝❡ ❢♦r♠✉❧❛ ✭❙t❡❢❛♥✐❝❛✱ ✷✵✶✶✱ ➓✶✳✶✵✮✱ Z (E)

t

ert(E−t) = ✶✱ s♦ t❤❛t Kt = e−rt(E−t)Ut =

Ut Z (E)

t

✇❤② ❆❚▼❋❄ ✾✶ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡

❉❡r✐✈❛t✐✈❡s✿ ❛ t✐♠❡ ❤♦♠♦❣❡♥❡♦✉s ❢r❛♠❡✇♦r❦

❛s ✇✐t❤ Z (E)

t

❢♦r ✜①❡❞ ✐♥❝♦♠❡✱ σ(K,E)

t

❝♦♥✈❡r❣❡s ❛s t → E ❝♦♥s✐❞❡r s❡t ♦❢ r♦❧❧✐♥❣ ✐♠♣❧✐❡❞ ♣❡r❝❡♥t❛❣❡ ✈♦❧❛t✐❧✐t✐❡s ✇✐t❤ s❛♠❡ t✐♠❡ t♦ ♠❛t✉r✐t② v✱ σ(Kt,t+v)

t

s✉❜st✐t✉t❡ ❆❚▼❋ ❞❡✜♥✐t✐♦♥ ❢♦r Kt ✐♥t♦ C BSM ♣r✐❝✐♥❣ ❢♦r♠✉❧❛ ❢♦r σ(Kt,E)

t

=

• ✽

E − t ❡r❢−✶

• C (Kt,E)

t

Ut

• ≈
• ✷π

v C (Kt,t+v)

t

Ut ❜② ✜rst ♦r❞❡r ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ ❡r❢−✶ ✭q✳✈✳ ❚❡❝❤♥✐❝❛❧ ❆♣♣❡♥❞✐① ➓✸✳✶✮ ♥♦r♠❛❧✐s❛t✐♦♥ ❜② Ut s❤♦✉❧❞ r❡♠♦✈❡ ♥♦♥✲st❛t✐♦♥❛r✐t② ♦❢ σ(Kt,E)

t

❛s C (Kt,t+v)

t

, Ut ♥♦t ✐♥✈❛r✐❛♥t✱ r❛t✐♦ ✉s✉❛❧❧② ♥♦t ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ♣✳✶✶✽✮✱ ❜✉t ❝❤❛♥❣❡s ✐♥ r♦❧❧✐♥❣ ❆❚▼❋ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ♣❛ss ❤❡✉r✐st✐❝ t❡sts ✭❧✐❦❡ ❞✐✛❡r❡♥❝✐♥❣ I (✶) s❡r✐❡s❄✮

✾✷ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t Pr♦❥❡❝t✐♥❣ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥

Pr♦❥❡❝t✐♥❣ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥

❤❛✈❡ ✐❞❡♥t✐✜❡❞ ✐♥✈❛r✐❛♥ts✱ ❳t,˜

τ ❣✐✈❡♥ ❡st✐♠❛t✐♦♥ ✐♥t❡r✈❛❧ ˜

τ ✇❛♥t t♦ ❦♥♦✇ ❞✐str✐❜✉t✐♦♥ ♦❢ ❳T+τ,τ✱ r✈ ❛t ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥✱ τ ♦✉r ♣r❡❢❡rr❡❞ ✐♥✈❛r✐❛♥ts ❛r❡ s♣❡❝✐✜❡❞ ✐♥ t❡r♠s ♦❢ ❞✐✛❡r❡♥❝❡s

✶

❝♦♠♣♦✉♥❞s r❡t✉r♥s ❢♦r ❡q✉✐t✐❡s✱ ❝♦♠♠♦❞✐t✐❡s✱ ❋❳ ❳T+τ,τ = ❧♥ PT+τ − ❧♥ PT

✷

❝❤❛♥❣❡s ✐♥ ❨❚▼ ❢♦r ✜①❡❞ ✐♥❝♦♠❡ ❳T+τ,τ = YT+τ − YT

✸

❝❤❛♥❣❡s ✐♥ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ❢♦r ❞❡r✐✈❛t✐✈❡s ❳T+τ,τ = σT+τ − σT

❛❧❧ ♦❢ ✇❤✐❝❤ ❛r❡ ❛❞❞✐t✐✈❡✱ s♦ t❤❛t t❤❡② s❛t✐s❢② ❳T+τ,τ = ❳T+τ,˜

τ + ❳T+τ−˜ τ,˜ τ + · · · + ❳T+˜ τ,˜ τ

✾✹ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t Pr♦❥❡❝t✐♥❣ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥

❉✐str✐❜✉t✐♦♥s ❛t t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥

❢♦r ❡①♣♦s✐t✐♦♥❛❧ s✐♠♣❧✐❝✐t②✱ ❛ss✉♠❡ t❤❛t τ = k˜ τ✱ ✇❤❡r❡ k ∈ Z++

♥♦ ♣r♦❜❧❡♠ ✐❢ ♥♦t ❛s ❧♦♥❣ ❛s ❞✐str✐❜✉t✐♦♥ ✐s ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ✭✇❤②❄✮

❛s ❛❧❧ ♦❢ t❤❡ ✐♥✈❛r✐❛♥ts ✐♥ ❳T+τ,τ = ❳T+τ,˜

τ + ❳T+τ−˜ τ,˜ τ + · · · + ❳T+˜ τ,˜ τ

❛r❡ ■■❉✱ t❤❡ ♣r♦❥❡❝t✐♦♥ ❢♦r♠✉❧❛ ✐s φ❳T+τ,τ =

• φ❳t,˜

τ

τ

˜ τ ♣r♦♦❢

❝❛♥ tr❛♥s❧❛t❡ ❜❛❝❦ ❛♥❞ ❢♦rt❤ ❜❡t✇❡❡♥ ❝❢ ❛♥❞ ♣❞❢ ✇✐t❤ ❋♦✉r✐❡r ❛♥❞ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠s φ❳ = F [f❳] ❛♥❞ f❳ = F−✶ [φ❳] ❜② ❝♦♥tr❛st✱ ❧✐♥❡❛r r❡t✉r♥ ♣r♦❥❡❝t✐♦♥s ②✐❡❧❞ ▲T+τ,τ = diag (✶ + ▲T+τ,˜

τ) × · · · × diag (✶ + ▲T+˜ τ,˜ τ) − ✶

✇❤❡r❡ t❤❡ ❞✐❛❣♦♥❛❧ ❡♥tr✐❡s ✐♥ t❤❡ N × N diag ♠❛tr✐① ❛r❡ t❤♦s❡ ✐♥ ✐ts ✈❡❝t♦r✲✈❛❧✉❡❞ ❛r❣✉♠❡♥t❀ ✐ts ♦✛✲❞✐❛❣♦♥❛❧ ❡♥tr✐❡s ❛r❡ ③❡r♦

✾✺ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t Pr♦❥❡❝t✐♥❣ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥

❏♦✐♥t ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥s

❊①❛♠♣❧❡

▲❡t t❤❡ ✇❡❡❦❧② ❝♦♠♣♦✉♥❞ r❡t✉r♥s ♦♥ ❛ st♦❝❦ ❛♥❞ t❤❡ ✇❡❡❦❧② ②✐❡❧❞ ❝❤❛♥❣❡s ❢♦r t❤r❡❡✲②❡❛r ❜♦♥❞s ❜❡ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞✳ ❚❤✉s✱ t❤❡ ✐♥✈❛r✐❛♥ts ❛r❡ ❳t,˜

τ =

• Ct,˜

τ

X (v)

t,˜ τ

• ≡
• ❧♥ Pt − ❧♥ Pt−˜

τ

Y (v)

t

− Y (v)

t−˜ τ

• .

❇✐♥❞ t❤❡s❡ ♠❛r❣✐♥❛❧s s♦ t❤❛t t❤❡✐r ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ✐s ❛❧s♦ ♥♦r♠❛❧✱ ❳t,˜

τ ∼ N (µ, Σ)✳ ❇② ❥♦✐♥t ♥♦r♠❛❧✐t②✱ t❤❡ ❝❢ ✐s φ❳t,˜

τ (ω) = eiω′µ− ✶ ✷ω′Σω✳

❋r♦♠ t❤❡ ♣r❡✈✐♦✉s s❧✐❞❡✱ ❳T+τ,τ ❤❛s ❝❢ φ❳T+τ,τ (ω) = eiω′ τ

˜ τ µ− ✶ ✷ω′ τ ˜ τ Σω✳

❚❤✉s✱ ❳T+τ,τ ∼ N τ ˜ τ µ, τ ˜ τ Σ

• .

✾✻ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t Pr♦❥❡❝t✐♥❣ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❤♦r✐③♦♥ ❞✐str✐❜✉t✐♦♥

t❤❡ ♣r♦❥❡❝t✐♦♥ ❢♦r♠✉❧❛ ❛❧❧♦✇s ❞❡r✐✈❛t✐♦♥ ♦❢ ♠♦♠❡♥ts ✭✇❤❡♥ t❤❡② ❛r❡ ❞❡✜♥❡❞✮

✶

❡①♣❡❝t❡❞ ✈❛❧✉❡s s✉♠ E {❳T+τ,τ} = τ ˜ τ E {❳t,˜

τ}

✷

sq✉❛r❡✲r♦♦t ♦❢ t✐♠❡ r✉❧❡ ♦❢ r✐s❦ ♣r♦♣❛❣❛t✐♦♥ Cov {❳T+τ,τ} = τ ˜ τ Cov {❳t,˜

τ} ⇔ Sd {❳T+τ,τ} =

τ ˜ τ Sd {❳t,˜

τ}

◆♦r♠❛❧✐s✐♥❣ ˜ τ = ✶ ②❡❛r✿ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ❤♦r✐③♦♥ ✐♥✈❛r✐❛♥t ✐s t❤❡ sq✉❛r❡ r♦♦t ♦❢ t❤❡ ❤♦r✐③♦♥ t✐♠❡s t❤❡ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ❛♥♥✉❛❧✐s❡❞ ✐♥✈❛r✐❛♥t

✐♥t✉✐t✐♦♥❄ P♦rt❢♦❧✐♦ ❞✐✈❡rs✐✜❡s ✐ts❡❧❢ ❜② r❡❝❡✐✈✐♥❣ ■■❉ s❤♦❝❦s ♦✈❡r t✐♠❡ s❡❡ ❉❛♥✐❡❧ss♦♥ ❛♥❞ ❩✐❣r❛♥❞ ✭✷✵✵✻✮ ❢♦r ✇❛r♥✐♥❣s ❛❜♦✉t ♥♦♥✲r♦❜✉st♥❡ss

✾✼ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ▼❛♣♣✐♥❣ ✐♥✈❛r✐❛♥ts ✐♥t♦ ♠❛r❦❡t ♣r✐❝❡s

❘❛✇ s❡❝✉r✐t✐❡s✿ ❤♦r✐③♦♥ ♣r✐❝❡s

♣r✐❝❡s ❞❡♣❡♥❞ ♦♥ ✐♥✈❛r✐❛♥ts t❤r♦✉❣❤ s♦♠❡ ♣r✐❝✐♥❣ ❢✉♥❝t✐♦♥✱ PT+τ = ❣ (❳T+τ,τ)

✶ ❢♦r ❡q✉✐t✐❡s✱ ♠❛♥✐♣✉❧❛t✐♥❣ t❤❡ ❝♦♠♣♦✉♥❞ r❡t✉r♥s ❢♦r♠✉❧❛ ②✐❡❧❞s

PT+τ = PTe❳T+τ,τ

✷ ❢♦r ③❡r♦ ❝♦✉♣♦♥ ❜♦✉♥❞s✱ ♠❛♥✐♣✉❧❛t✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ ❘(E−T−τ)

T+τ,τ

❛♥❞ ❳ (E−T−τ)

T+τ,τ

②✐❡❧❞s ❩ (E)

T+τ = ❩ (E−τ) T

e−(E−T−τ)❳ (E−T−τ)

T+τ,τ

♥✳❜✳ ❝♦✉❧❞ ✉s❡ v ≡ E − (T + τ)

✾✾ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ▼❛♣♣✐♥❣ ✐♥✈❛r✐❛♥ts ✐♥t♦ ♠❛r❦❡t ♣r✐❝❡s

❘❛✇ s❡❝✉r✐t✐❡s✿ ❤♦r✐③♦♥ ♣r✐❝❡ ❞✐str✐❜✉t✐♦♥

❢♦r ❜♦t❤ ❡q✉✐t✐❡s ❛♥❞ ✜①❡❞ ✐♥❝♦♠❡✱ PT+τ = e❨T+τ,τ ✱ ✇❤❡r❡ ❨T+τ,τ ≡ γ + diag (ε) ❳T+τ,τ ❛♥ ❛✣♥❡ tr❛♥s❢♦r♠❛t✐♦♥ t❤✉s✱ t❤❡② ❤❛✈❡ ❛ ❧♦❣ −❨ ❞✐str✐❜✉t✐♦♥ t❤✐s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s φ❨T+τ,τ (ω) = eiω′γφ❳T+τ,τ (diag (ε) ω) ✉s✉❛❧❧② ✐♠♣♦ss✐❜❧❡ t♦ ❝♦♠♣✉t❡ ❝❧♦s❡❞ ❢♦r♠ ❢♦r ❢✉❧❧ ❞✐str✐❜✉t✐♦♥ ♠❛② s✉✣❝❡ ❥✉st t♦ ❝♦♠♣✉t❡ ✜rst ❢❡✇ ♠♦♠❡♥ts ❡✳❣✳ ❝❛♥ ❝♦♠♣✉t❡ E {Pn} ❛♥❞ Cov {Pm, Pn} ❢r♦♠ ❝❢

✶✵✵ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ▼❛♣♣✐♥❣ ✐♥✈❛r✐❛♥ts ✐♥t♦ ♠❛r❦❡t ♣r✐❝❡s

❉❡r✐✈❛t✐✈❡s✿ ❤♦r✐③♦♥ ♣r✐❝❡s

♣r✐❝❡s ❛r❡ st✐❧❧ ❢✉♥❝t✐♦♥s ♦❢ ✐♥✈❛r✐❛♥ts✱ PT+τ = ❣ (❳T+τ,τ) ❛s ♣r✐❝❡s r❡✢❡❝t ♠✉❧t✐♣❧❡ ✐♥✈❛r✐❛♥ts✱ ♥♦ ❧♦♥❣❡r s✐♠♣❧❡ ❧♦❣ −❨ str✉❝t✉r❡

❊①❛♠♣❧❡

❆❣❛✐♥✿ ♣r✐❝❡ ♦❢ ❛ ❊✉r♦♣❡❛♥ ❝❛❧❧ ♦♣t✐♦♥ ❛t ❤♦r✐③♦♥ T + τ ≤ E ✐s C (K,E)

T+τ

≡ C BSM E − T − τ, K, UT+τ, Z (E)

T+τ, σ(K,E) T+τ

• .

❚❤❡ ❤♦r✐③♦♥ ❞✐str✐❜✉t✐♦♥s ♦❢ t❤❡ t❤r❡❡ ✐♥✈❛r✐❛♥ts ❛r❡ t❤❡♥ UT+τ = UTeX✶ Z (E)

T+τ = Z (E−τ) T

e−X✷v σ(K,E)

T+τ = σ(KT ,E−τ) T

+ X✸ ❢♦r v ≡ E − T − τ ❛♥❞ s✉✐t❛❜❧② ❞❡✜♥❡❞ KT ❛♥❞ ✐♥✈❛r✐❛♥ts✱ X✶ t♦ X✸✳

✶✵✶ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ▼❛♣♣✐♥❣ ✐♥✈❛r✐❛♥ts ✐♥t♦ ♠❛r❦❡t ♣r✐❝❡s

❉❡r✐✈❛t✐✈❡s✿ ❛♣♣r♦①✐♠❛t✐♥❣ ❤♦r✐③♦♥ ♣r✐❝❡s

♦♣t✐♦♥s ♣r✐❝✐♥❣ ❢♦r♠✉❧❛ ✐s ❛❧r❡❛❞② ❝♦♠♣❧✐❝❛t❡❞✱ ♥♦♥✲❧✐♥❡❛r ❛❞❞✐♥❣ ✐♥ ♣♦ss✐❜❧② ❝♦♠♣❧✐❝❛t❡❞ ❤♦r✐③♦♥ ♣r♦❥❡❝t✐♦♥s ❛❧♠♦st ❝❡rt❛✐♥❧② ♣r❡✈❡♥ts ❡①❛❝t s♦❧✉t✐♦♥s ❜✉t ❝❛♥ ❛♣♣r♦①✐♠❛t❡ PT+τ = g (❳T+τ,τ) ✇✐t❤ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ PT+τ ≈ g (♠) + (❳ − ♠) ∇g (♠) + ✶ ✷ (❳ − ♠)′ H (g (♠)) (❳ − ♠) ✇❤❡r❡ ∇g (♠) ✐s ❣r❛❞✐❡♥t✱ H (g (♠)) ❍❡ss✐❛♥ ❛♥❞ ♠ s♦♠❡ s✐❣♥✐✜❝❛♥t ✈❛❧✉❡ ♦❢ t❤❡ ✐♥✈❛r✐❛♥ts ❳T+τ,τ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦❞✉❝❡s t❤❡ ●r❡❡❦s

❊①❛♠♣❧❡ ✭❇❡t❖♥▼❛r❦❡ts✮

❇❡t❖♥▼❛r❦❡ts ❤❛s t♦ ♣r✐❝❡ ❝✉st♦♠ ♦♣t✐♦♥s ✐♥ ❧❡ss t❤❛♥ ✶✺ s❡❝♦♥❞s✳ ▼♦♥t❡ ❈❛r❧♦ ✐s ❢❛r t♦♦ s❧♦✇❀ ❡✈❡♥ ❇❧❛❝❦✲❙❝❤♦❧❡s ♠❛② ❜❡✳ ❚❤❡② ✉s❡ ❱❛♥♥❛✲❱♦❧❣❛✳

✶✵✷ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ▼❛♣♣✐♥❣ ✐♥✈❛r✐❛♥ts ✐♥t♦ ♠❛r❦❡t ♣r✐❝❡s

▲❡❝t✉r❡ ✺ ❡①❡r❝✐s❡s

▼❡✉❝❝✐ ❡①❡r❝✐s❡s

♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✺✳✸ ▼❆❚▲❆❇✿ ✸✳✷✳✷✱ ✸✳✷✳✸✱ ✺✳✶ ✭♠♦❞✐❢② ❝♦❞❡ t♦ ❞✐s♣❧❛② ♦♥❡✲♣❡r✐♦❞ ❛♥❞ ❤♦r✐③♦♥ ❞✐str✐❜✉t✐♦♥s❀ ❝♦♥tr❛st t♦ ▼❡✉❝❝✐ ✭✷✵✵✺✮ ❡q✉❛t✐♦♥s ✸✳✾✺✱ ✸✳✾✻✮✱ ✺✳✺✳✶✱ ✺✳✺✳✷✱ ✺✳✻

♣r♦❥❡❝t

♣r♦❞✉❝❡ ❤♦r✐③♦♥ ♣r✐❝❡ ❞✐str✐❜✉t✐♦♥s ❢♦r ②♦✉r ❛ss❡ts

✶✵✸ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

❲❤② ❞✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥❄

✶ ❛❝t✉❛❧ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ♠❛r❦❡t ✐s ❧❡ss t❤❛♥ t❤❡ ♥✉♠❜❡r ♦❢ s❡❝✉r✐t✐❡s

❊①❛♠♣❧❡

❈♦♥s✐❞❡r ❛ st♦❝❦ ✇❤♦s❡ ♣r✐❝❡ ✐s Ut ❛♥❞ ❛ ❊✉r♦♣❡❛♥ ❝❛❧❧ ♦♣t✐♦♥ ♦♥ ✐t ✇✐t❤ str✐❦❡ K ❛♥❞ ❡①♣✐r② ❞❛t❡ T + τ✳ ❚❤❡✐r ❤♦r✐③♦♥ ♣r✐❝❡s ❛r❡ PT+τ = UT+τ ♠❛① {UT+τ − K, ✵}

• .

❚❤❡s❡ ❛r❡ ♣❡r❢❡❝t❧② ♣♦s✐t✐✈❡❧② ❞❡♣❡♥❞❡♥t✳

✷ ❛❝t✉❛❧ r❛♥❞♦♠♥❡ss ✐♥ t❤❡ ♠❛r❦❡t ❝❛♥ ❜❡ ✇❡❧❧ ❛♣♣r♦①✐♠❛t❡❞ ✇✐t❤ ❢❡✇❡r

t❤❛♥ N ❞✐♠❡♥s✐♦♥s ✭t❤❛t ♦❢ t❤❡ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts✱ ❳✮

t❤✐s ✐s t❤❡ ♣♦ss✐❜✐❧✐t② ❝♦♥s✐❞❡r❡❞ ✐♥ ✇❤❛t ❢♦❧❧♦✇s ❝❛♥ ❝♦♥s✐❞❡r❛❜❧② r❡❞✉❝❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t②

✶✵✺ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

❈♦♠♠♦♥ ❢❛❝t♦rs

✇♦✉❧❞ ❧✐❦❡ t♦ ❡①♣r❡ss N✲✈❡❝t♦r ❳t,˜

τ ✐♥ t❡r♠s ♦❢

✶

❛ K✲✈❡❝t♦r ♦❢ ❝♦♠♠♦♥ ❢❛❝t♦rs✱ ❋t,˜

τ❀

✶

❡①♣❧✐❝✐t ❢❛❝t♦rs ❛r❡ ♠❡❛s✉r❛❜❧❡ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts

✷

❤✐❞❞❡♥ ❢❛❝t♦rs ❛r❡ s②♥t❤❡t✐❝ ✐♥✈❛r✐❛♥ts ❡①tr❛❝t❡❞ ❢r♦♠ t❤❡ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts

✷

❛♥ N✲✈❡❝t♦r ♦❢ r❡s✐❞✉❛❧ ♣❡rt✉r❜❛t✐♦♥s✱ ❯t,˜

τ

❛s ❢♦❧❧♦✇s ❳t,˜

τ = ❤ (❋t,˜ τ) + ❯t,˜ τ

❢♦r tr❛❝t❛❜✐❧✐t②✱ ✉s✉❛❧❧② ✉s❡ ❧✐♥❡❛r ❢❛❝t♦r ♠♦❞❡❧ ✭✜rst ♦r❞❡r ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥✮✱ ❳t,˜

τ = ❇❋t,˜ τ + ❯t,˜ τ

✇✐t❤ ❛♥ N × K ❢❛❝t♦r ❧♦❛❞✐♥❣ ♠❛tr✐①✱ ❇

✶✵✻ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

❈♦♠♠♦♥ ❢❛❝t♦rs✿ ❞❡s✐❞❡r❛t❛

✶ s✉❜st❛♥t✐❛❧ ❞✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥✱ K ≪ N ✷ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ❋t,˜

τ ❛♥❞ ❯t,˜ τ ✭✇❤②❄✮

❤❛r❞ t♦ ❛tt❛✐♥✱ s♦ ♦❢t❡♥ r❡❧❛① t♦ Cor {❋t,˜

τ, ❯t,˜ τ} = ✵K×N

✸ ❣♦♦❞♥❡ss ♦❢ ✜t

✇❛♥t r❡❝♦✈❡r❡❞ ✐♥✈❛r✐❛♥ts t♦ ❜❡ ❝❧♦s❡✱ ˜ ❳ ≡ ❤ (❋) ≈ ❳ ✉s❡ ❣❡♥❡r❛❧✐s❡❞ R✷ R✷ ❳, ˜ ❳

• ≡ ✶ −

E

• ❳ − ˜

❳ ′ ❳ − ˜ ❳

• tr {Cov {❳}}

✇❤❡r❡ t❤❡ tr❛❝❡ ♦❢ ❨ ✱ tr {❨ }✱ ✐s t❤❡ s✉♠ ♦❢ ✐ts ❞✐❛❣♦♥❛❧ ❡♥tr✐❡s

✶

✇❤❛t ✐s ✐♥ t❤❡ ♥✉♠❡r❛t♦r❄

✷

✇❤❛t ✐s ✐♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r❄

✸

❤♦✇ ❞♦❡s t❤✐s ❞✐✛❡r ❢r♦♠ t❤❡ ✉s✉❛❧ ❝♦❡✣❝✐❡♥t ♦❢ ❞❡t❡r♠✐♥❛t✐♦♥✱ R✷❄

✶✵✼ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

❊①♣❧✐❝✐t ❢❛❝t♦rs

s✉♣♣♦s❡ t❤❛t t❤❡♦r② ♣r♦✈✐❞❡s ❛ ❧✐st ♦❢ ❡①♣❧✐❝✐t ♠❛r❦❡t ✈❛r✐❛❜❧❡s ❛s ❢❛❝t♦rs✱ ❋ ❤♦✇ ❞♦❡s ♦♥❡ ❞❡t❡r♠✐♥❡ t❤❡ ❧♦❛❞✐♥❣s ♠❛tr✐①✱ ❇❄ ✇✐t❤ ❧✐♥❡❛r ❢❛❝t♦r ♠♦❞❡❧✱ ❳ = ❇❋ + ❯✱ ♣✐❝❦ ❇ t♦ ♠❛①✐♠✐s❡ ❣❡♥❡r❛❧✐s❡❞ R✷ ❇r ≡ ❛r❣♠❛①

❇

R✷ {❳, ❇❋} ✇❤❡r❡ t❤❡ s✉❜s❝r✐♣t ✐♥❞✐❝❛t❡s t❤❛t t❤❡s❡ ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② r❡❣r❡ss✐♦♥ t❤✐s ✐s s♦❧✈❡❞ ❜② ❇r = E

• ❳❋ ′

E

• ❋❋ ′−✶

❤♦✇ ❞♦❡s t❤✐s ❞✐✛❡r ❢r♦♠ ❖▲❙❄

❡✈❡♥ ✇❡❛❦ ✈❡rs✐♦♥ ♦❢ s❡❝♦♥❞ ❞❡s✐❞❡r❛t✉♠✱ Cor {❋, ❯} = ✵K×N ♥♦t ❣❡♥❡r❛❧❧② s❛t✐s✜❡❞❀ ❜✉t✿

✶

E {❋} = ✵ ⇒ Cor {❋, ❯} = ✵K×N

✷

❛❞❞✐♥❣ ❝♦♥st❛♥t ❢❛❝t♦r t♦ ❋ ⇒ E {❯r} = ✵, Cor {❋, ❯r} = ✵K×N

❝❢✳ ✐♥❝❧✉❞✐♥❣ ❝♦♥st❛♥t t❡r♠ ✐♥ ❖▲❙ r❡❣r❡ss✐♦♥

✶✵✽ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

❊①♣❧✐❝✐t ❢❛❝t♦rs✿ ♣✐❝❦✐♥❣ ❢❛❝t♦rs

✶ ✇❛♥t t❤❡ s❡t ♦❢ ❢❛❝t♦rs t♦ ❜❡ ❛s ❤✐❣❤❧② ❝♦rr❡❧❛t❡❞ ❛s ♣♦ss✐❜❧❡ ✇✐t❤ t❤❡

♠❛r❦❡t ✐♥✈❛r✐❛♥ts

♠❛①✐♠✐s❡s ❡①♣❧❛♥❛t♦r② ♣♦✇❡r ♦❢ t❤❡ ❢❛❝t♦rs ✐❢ ❞♦ ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts ❞❡❝♦♠♣♦s✐t✐♦♥ ♦♥ ❋✱ s♦ t❤❛t Cov {❋} = ❊Λ❊ ′ ❛♥❞ ❈❳❋ ≡ Cor {❳, ❊ ′❋} ✭❊ ′❋ ❛r❡ r♦t❛t❡❞ ❢❛❝t♦rs✮ t❤❡♥ R✷ ❳, ˜ ❳r

• = tr (❈❳❋❈ ′

❳❋)

N

✷ ✇❛♥t t❤❡ s❡t ♦❢ ❢❛❝t♦rs t♦ ❜❡ ❛s ✉♥❝♦rr❡❧❛t❡❞ ✇✐t❤ ❡❛❝❤ ♦t❤❡r ❛s

♣♦ss✐❜❧❡

❡①tr❡♠❡ ✈❡rs✐♦♥ ♦❢ ❝♦rr❡❧❛t✐♦♥ ✐s ♠✉❧t✐❝♦❧❧✐♥❡❛r✐t② ✐♥ t❤✐s ❝❛s❡✱ ❛❞❞✐♥❣ ❛❞❞✐t✐♦♥❛❧ ❢❛❝t♦rs ❞♦❡s♥✬t ❛❞❞ ❡①♣❧❛♥❛t♦r② ♣♦✇❡r✱ ❛♥❞ ❧❡❛✈❡s r❡❣r❡ss✐♦♥ ♣❧❛♥❡ ✐❧❧ ❝♦♥❞✐t✐♦♥❡❞

✸ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ tr❛❞❡✲♦✛ ❜❡t✇❡❡♥ ♠♦r❡ ❛❝❝✉r❛❝② ❛♥❞ ♠♦r❡

❝♦♠♣✉t❛t✐♦♥❛❧ ✐♥t❡♥s✐✈✐t② ✇❤❡♥ ❛❞❞✐♥❣ ❢❛❝t♦rs

✶✵✾ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

❊①❛♠♣❧❡ ✭❈❛♣✐t❛❧ ❛ss❡ts ♣r✐❝✐♥❣ ♠♦❞❡❧ ✭❈❆P▼✮✮

❚❤❡ ❧✐♥❡❛r r❡t✉r♥s ✭✐♥✈❛r✐❛♥ts✮ ♦❢ N st♦❝❦s ❛r❡ L(n)

t,˜ τ ≡ P(n)

t

P(n)

t−˜ τ

− ✶✳ ■❢ t❤❡ ♣r✐❝❡ ♦❢ t❤❡ ♠❛r❦❡t ✐♥❞❡① ✐s Mt✱ t❤❡ ❧✐♥❡❛r r❡t✉r♥ ♦♥ t❤❡ ♠❛r❦❡t ✐♥❞❡①✱ F M

t,˜ τ ≡ Mt Mt−˜

τ − ✶✱ ✐s ❛ ❧✐♥❡❛r ❢❛❝t♦r✳ ❘❡❣r❡ss✐♦♥ t❤❡♥ ❡st✐♠❛t❡s β(n)

˜ τ ✱ ❛

❤♦r✐③♦♥✲❞❡♣❡♥❞❡♥t ❝♦rr❡❧❛t✐♦♥✿ ˜ L(n)

t,˜ τ = E

• L(n)

t,˜ τ

• + β(n)

˜ τ

• F M

t,˜ τ − E

• F M

t,˜ τ

• .

■❢ ❡①♣❡❝t❡❞ r❡t✉r♥s ❝♦♥✈❡①❧② ❝♦♠❜✐♥❡ ♠❛r❦❡t ❛♥❞ r✐s❦✲❢r❡❡ r❡t✉r♥s✱ E

• L(n)

t,˜ τ

• = β(n)

˜ τ E

• F M

t,˜ τ

• +
• ✶ − β(n)

˜ τ

• Rf

t,˜ τ

t❤❡♥ t❤❡ ❈❆P▼ ❢♦❧❧♦✇s ˜ L(n)

t,˜ τ = Rf t,˜ τ + β(n) ˜ τ

• F M

t,˜ τ − Rf t,˜ τ

• .

✶✶✵ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

❊①❛♠♣❧❡ ✭❋❛♠❛ ❛♥❞ ❋r❡♥❝❤ ✭✶✾✾✸✮ t❤r❡❡ ❢❛❝t♦r ♠♦❞❡❧✮

❚❤❡ ❋❛♠❛ ❛♥❞ ❋r❡♥❝❤ ✭✶✾✾✸✮ t❤r❡❡ ❢❛❝t♦r ♠♦❞❡❧ r❡❞✉❝❡s t❤❡ ❝♦♠♣♦✉♥❞ r❡t✉r♥s✱ C (n)

t,˜ τ ♦❢ N st♦❝❦s t♦ t❤r❡❡ ❡①♣❧✐❝✐t ❧✐♥❡❛r ❢❛❝t♦rs ❛♥❞ ❛ ❝♦♥st❛♥t✿

✶ C M✱ t❤❡ ❝♦♠♣♦✉♥❞ r❡t✉r♥ t♦ ❛ ❜r♦❛❞ st♦❝❦ ✐♥❞❡① ✷ SmB✱ s♠❛❧❧ ♠✐♥✉s ❜✐❣✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝♦♠♣♦✉♥❞ r❡t✉r♥ t♦

❛ s♠❛❧❧✲❝❛♣ st♦❝❦ ✐♥❞❡① ❛♥❞ ❛ ❧❛r❣❡✲❝❛♣ st♦❝❦ ✐♥❞❡①

✸ HmL✱ ❤✐❣❤ ♠✐♥✉s ❧♦✇✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝♦♠♣♦✉♥❞ r❡t✉r♥ t♦

❛ ❤✐❣❤ ❜♦♦❦✲t♦✲♠❛r❦❡t st♦❝❦ ✐♥❞❡① ❛♥❞ ❛ ❧♦✇ ❜♦♦❦✲t♦✲♠❛r❦❡t st♦❝❦ ✐♥❞❡① ❯♥❧✐❦❡ ♠♦❞❡❧s ♦❢ ❧✐♥❡❛r r❡t✉r♥s✱ t❤❡ r❡❣r❡ss✐♦♥ ❝♦❡✣❝✐❡♥ts ❤❡r❡ ❞❡♣❡♥❞ ♦♥ ❝♦✈❛r✐❛♥❝❡s❀ ❜② ❵sq✉❛r❡ r♦♦t✬ ♣r♦♣❡rt②✱ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡st✐♠❛t✐♦♥ ✐♥t❡r✈❛❧✱ ˜ τ

✶✶✶ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

❍✐❞❞❡♥ ❢❛❝t♦rs

♥♦✇ ❧❡t ❢❛❝t♦rs✱ ❋ (❳t,˜

τ) ❜❡ s②♥t❤❡t✐❝ ✐♥✈❛r✐❛♥ts ❡①tr❛❝t❡❞ ❢r♦♠ ♠❛r❦❡t

✐♥✈❛r✐❛♥ts t❤✉s✱ t❤❡ ❛✣♥❡ ♠♦❞❡❧ ✐s ❳t,˜

τ = q + ❇❋ (❳t,˜ τ) + ❯t,˜ τ

✇❤❛t ✐s t❤❡ tr✐✈✐❛❧ ✇❛② ♦❢ ♠❛①✐♠✐s✐♥❣ ❣❡♥❡r❛❧✐s❡❞ R✷❄

✇❤❛t ✐s t❤❡ ✇❡❛❦♥❡ss ♦❢ ❞♦✐♥❣ s♦❄

♦t❤❡r✇✐s❡✱ ♠❛✐♥ ❛♣♣r♦❛❝❤ t❛❦❡♥ ✐s ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥t ❛♥❛❧②s✐s ✭P❈❆✮

✶✶✷ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

Pr✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥t ❛♥❛❧②s✐s ✭P❈❆✮

❛ss✉♠❡ t❤❡ ❤✐❞❞❡♥ ❢❛❝t♦rs ❛r❡ ❛✣♥❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❳t,˜

τ

❋p (❳t,˜

τ) = ❞p + ❆′ p❳t,˜ τ

❣✐✈❡♥ t❤❡s❡ ❛✣♥❡ ❛ss✉♠♣t✐♦♥s✱ t❤❡ ♦♣t✐♠❛❧❧② r❡❝♦✈❡r❡❞ ✐♥✈❛r✐❛♥ts ❛r❡ ˜ ❳p = ♠p + ❇p❆′

p❳t,˜ τ

✇❤❡r❡ (❇p, ❆p, ♠p) ≡ ❛r❣♠❛①

❇,❆,♠

R✷ ❳, ♠ + ❇❆′❳t,˜

τ

• ❤❡✉r✐st✐❝❛❧❧②

✇❛♥t ♦rt❤♦❣♦♥❛❧ ❢❛❝t♦rs ❝♦♥s✐❞❡r ❧♦❝❛t✐♦♥✲❞✐s♣❡rs✐♦♥ ❡❧❧✐♣s♦✐❞ ❣❡♥❡r❛t❡❞ ❜② ❳t,˜

τ

❛s❦✐♥❣ ✇❤❛t ✐ts ❧♦♥❣❡st ♣r✐♥❝✐♣❛❧ ❛①❡s ❛r❡

✶✶✸ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

▲♦❝❛t✐♦♥✲❞✐s♣❡rs✐♦♥ ❡❧❧✐♣s♦✐❞

❝♦♥s✐❞❡r r✈ ❳ ✐♥ R✸ ❣✐✈❡♥ ❧♦❝❛t✐♦♥ ❛♥❞ ❞✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡rs✱ µ ❛♥❞ Σ✱ ❝❛♥ ❢♦r♠ ❧♦❝❛t✐♦♥✲❞✐s♣❡rs✐♦♥ ❡❧❧✐♣s♦✐❞ ✐❢ K = ✶✱ ✇❤✐❝❤ ❢❛❝t♦r ✇♦✉❧❞ ②♦✉ ❝❤♦♦s❡❄ ❲❤❛t ✇♦✉❧❞ ˜ ❳p ❧♦♦❦ ❧✐❦❡❄ ✇❤❛t ✐❢ K = ✷❄ ✇❤❛t ✐❢ K = ✸❄

✶✶✹ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

❖♣t✐♠❛❧ ❢❛❝t♦rs ✐♥ P❈❆

❝❤♦♦s✐♥❣ ♦♣t✐♠❛❧ ❢❛❝t♦rs t❤❡r❡❢♦r❡ r♦t❛t❡s✱ tr❛♥s❧❛t❡s ❛♥❞ ❝♦❧❧❛♣s❡s t❤❡ ❧♦❝❛t✐♦♥✲❞✐s♣❡rs✐♦♥ ❡❧❧✐♣s♦✐❞✬s ❝♦✲♦r❞✐♥❛t❡s ✭q✳✈✳ ▼❡✉❝❝✐✱ ✷✵✵✺✱ ❆♣♣ ❆✳✺✮ t❤✉s (❇p, ❆p, ♠p) =

• ❊K, ❊K,
• ■N − ❊K❊ ′

K

• E {❳t,˜

τ}

• ✇❤❡r❡

❊K ≡

• ❡(✶), . . . , ❡(K)

✇✐t❤ ❡(k) ❜❡✐♥❣ t❤❡ ❡✐❣❡♥✈❡❝t♦r ♦❢ Cov {❳t,˜

τ} ❝♦rr❡s♣♦♥❞✐♥❣ t♦ λk✱

t❤❡ kt❤ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡✳ ♠p tr❛♥s❧❛t❡s✱ ❛♥❞ ❇p❆′

p r♦t❛t❡s ❛♥❞ ❝♦❧❧❛♣s❡s✱ ❢♦r

˜ ❳p = ♠p + ❇p❆′

p❳t,˜ τ =

• ■N − ❊K❊ ′

K

• E {❳t,˜

τ} + ❊K❊ ′ K❳t,˜ τ

=E {❳t,˜

τ} + ❊K❊ ′ K (❳t,˜ τ − E {❳t,˜ τ})

✇❤② ❛r❡ E {❯p} = ✵ ❛♥❞ Cor {❋p, ❯p} = ✵K×N❄ ❛s R✷ ❳t,˜

τ, ˜

❳p

• =

K

k=✶ λk

N

n=✶ λn ✱ ❝❛♥ s❡❡ ❡✛❡❝t ♦❢ ❡❛❝❤ ❢✉rt❤❡r ❢❛❝t♦r✶✶✺ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥

❊①♣❧✐❝✐t ❢❛❝t♦rs ✈ P❈❆❄

❛s P❈❆ ♣r♦❥❡❝ts ♦♥t♦ t❤❡ ♠♦st ✐♥❢♦r♠❛t✐✈❡ K ❞✐♠❡♥s✐♦♥s✱ ✐t ②✐❡❧❞s ❛ ❤✐❣❤❡r R✷ t❤❛♥ ❛♥② K✲❢❛❝t♦r ❡①♣❧✐❝✐t ❢❛❝t♦r ♠♦❞❡❧ ❤♦✇❡✈❡r✱ t❤❡ s②♥t❤❡t✐❝ ❞✐♠❡♥s✐♦♥s ♦❢ P❈❆ ❛r❡ ❤❛r❞❡r t♦ ✐♥t❡r♣r❡t✱ ❛♥❞ t❤❡r❡❢♦r❡ ♣❡r❤❛♣s t♦ ✉♥❞❡rst❛♥❞

❜✉t s❡❡ ▼❡✉❝❝✐ ✭✷✵✵✺✱ ♣♣✳✶✺✼✲✮ ❢♦r ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ s✇❛♣ ♠❛r❦❡t ②✐❡❧❞ ❝✉r✈❡ ✐♥t♦ ❧❡✈❡❧✱ s❧♦♣❡ ❛♥❞ ❝✉r✈❛t✉r❡ ❢❛❝t♦rs

s❡❡ ♣♣✳✻✼✲ ♦❢ ❙♠✐t❤ ❛♥❞ ❋✉❡rt❡s✬ P❛♥❡❧ ❚✐♠❡ ❙❡r✐❡s ♥♦t❡s ❢♦r ❛ ❞✐s❝✉ss✐♦♥ ♦❢ ❤♦✇ t♦ ✉s❡ ❛♥❞ ✐♥t❡r♣r❡t P❈❆ ♠♦❞❡❧s

✶✶✻ ✴ ✷✷✹

❊st✐♠❛t✐♥❣ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts

❊st✐♠❛t✐♥❣ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts✿ ❛ ❝❛♥♥❡❞ s✉♠♠❛r②

✇❡ ❧❛r❣❡❧② ❧❡❛✈❡ ❡st✐♠❛t✐♦♥ t♦ t❤❡ ❡❝♦♥♦♠❡tr✐❝s ♠♦❞✉❧❡❀ ❤♦✇❡✈❡r ✳ ✳ ✳ ❛♥ ❡st✐♠❛t♦r✿ ❛ ❢✉♥❝t✐♦♥ ♠❛♣♣✐♥❣ ❢r♦♠ ✐T t♦ ❛ ♥✉♠❜❡r✱ t❤❡ ❡st✐♠❛t❡ ✐ts ❡rr♦r ✏✐s t❤❡ ❛✈❡r❛❣❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❬❡st✐♠❛t❡❪ ❛♥❞ t❤❡ tr✉❡ ✈❛❧✉❡ ✳ ✳ ✳ ❡st✐♠❛t❡❞ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ s❝❡♥❛r✐♦s✑✿ Err✷ ˆ

• , ●
• ≡ E
• ˆ
• (■T) − ● [f❳] ✷

= Bias✷ ˆ

• , ●
• + Inef ✷

ˆ

• ✇❤❡r❡ ● [f❳] ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ tr✉❡ ❞✐str✐❜✉t✐♦♥✱ ˆ
• [✐T] ✐s ✐ts

❡st✐♠❛t❡✱ ❜✐❛s ✐s ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ E

• ˆ
• [■t]
• ❛♥❞ t❤❡ ✉♥❦♥♦✇♥
• [f❳]✱ ❛♥❞ ✐♥❡✣❝✐❡♥❝② ✐s t❤❡ ❡st✐♠❛t♦r✬s ❞✐s♣❡rs✐♦♥

♥♦♥✲♣❛r❛♠❡tr✐❝ ❡st✐♠❛t♦rs ✐♠♣♦s❡ ♥♦ ✐❞❡♥t✐❢②✐♥❣ r❡str✐❝t✐♦♥s ♦♥ t❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥❀ ❣♦♦❞ ❡st✐♠❛t❡s✱ t❤❡♥✱ r❡q✉✐r❡ ❧❛r❣❡ s❛♠♣❧❡s ♣❛r❛♠❡tr✐❝ ❡st✐♠❛t♦rs r❡str✐❝t ❞✐str✐❜✉t✐♦♥s✱ s♦ ❝❛♥ ❡st✐♠❛t❡ ✇❡❧❧ ♦♥ s♠❛❧❧ s❛♠♣❧❡s ✭✉♥❧❡ss ❜❛❞ ♣❛r❛♠❡tr✐❝ r❡str✐❝t✐♦♥s ❤❛✈❡ ❜❡❡♥ ♠❛❞❡✮ ❢♦r t❤❡ s♠❛❧❧❡st s❛♠♣❧❡s✱ s❤r✐♥❦❛❣❡ ❡st✐♠❛t♦rs ♣❡r❢♦r♠ ❇❛②❡s✐❛♥ ❛✈❡r❛❣❡s ♦❢ ❡st✐♠❛t❡❞ ✈❛❧✉❡s ✇✐t❤ ❛ ❝♦♥st❛♥t✱ r❡❞✉❝✐♥❣ ❡rr♦r ❜② ✐♠♣r♦✈✐♥❣ ❡✣❝✐❡♥❝② ❛t t❤❡ ❝♦st ♦❢ ❜✐❛s

❇❛②❡s✲❙t❡✐♥ s❛♠♣❧❡✲❜❛s❡❞ ❛❧❧♦❝❛t✐♦♥ ✶✶✼ ✴ ✷✷✹

❊st✐♠❛t✐♥❣ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts

❊①❛♠♣❧❡ ✭❊st✐♠❛t✐♥❣ t❤❡ ♠❡❛♥ ✐♥ s♠❛❧❧ s❛♠♣❧❡s✮

▲❡t X ❜❡ ❣❡♥❡r❛t❡❞ ❜② ✐♥❞❡♣❡♥❞❡♥t t❤r♦✇s ♦❢ ❛ ❢❛✐r ❞✐❡ s✉❝❤ t❤❛t E {X} = µ = ✼

✷✳ ❙✉♣♣♦s❡ ✇❡ ❞♦ ♥♦t ❦♥♦✇ µ✱ ❜✉t ✇✐s❤ t♦ ❡st✐♠❛t❡ ✐t✳

❈♦♥s✐❞❡r t❤❡ s❛♠♣❧❡ ♠❡❛♥✱ ˆ s ≡ ✶

T

T

t=✶ xt✳ ❚❤✐s ✐s ✉♥❜✐❛s❡❞✱

E {ˆ s − µ} = ✵✳ ❍♦✇❡✈❡r✱ ✐t ♠❛② ❜❡ ✐♥❡✣❝✐❡♥t ❛s Var {ˆ s} ≡ E

• (ˆ

s − µ)✷ ♠❛② ❜❡ ❤✐❣❤✳ ❲❤❡♥ T = ✶✱ ❢♦r ❡①❛♠♣❧❡✱ Var {ˆ s} = ✶ ✻ ✷✺ ✹ + ✾ ✹ + ✶ ✹

• × ✷ = ✸✺

✶✷ < ✸. ❲❤❡♥ T = ✷✱ Var {ˆ s} = · · · = ✶✵✺ ✼✷ < ✸✺ ✶✷. ❚❤✉s✱ ✇❤❡♥ T = ✷✱ ˆ s ❤❛s ❛♥ ❡rr♦r ♦❢

• ✵ + ✶✵✺

✼✷ =

• ✸✺

✷✹✳ ❲❤❡♥ T = ✶✱ t❤❡

❡rr♦r ✐s √ ✸✳ ◆♦✇ ❝♦♥s✐❞❡r t❤❡ ✜①❡❞ ❡st✐♠❛t♦r ˜ s ≡ ✷✳ ■ts ❜✐❛s ✐s ✸

✷✱ ❜✉t ✐ts ✈❛r✐❛♥❝❡ ✐s

③❡r♦✱ ❢♦r ❛♥ ❡rr♦r ♦❢ ✸

✷ ✕ ❜❡tt❡r t❤❛♥ ˆ

s ❢♦r T = ✶✱ ❜✉t ✇♦rs❡ ✇❤❡♥ T ≥ ✷✳

✶✶✽ ✴ ✷✷✹

❊st✐♠❛t✐♥❣ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts

❲❡✐❣❤t❡❞ ❡st✐♠❛t❡s

✐❢ iT ≡ {①✶, . . . , ①T} tr✉❧② ❣❡♥❡r❛t❡❞ ❜② ■■❉ ✐♥✈❛r✐❛♥ts✱ t❤❡♥ ❝❛♥ ✇♦r❦ ✇✐t❤ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥s✱ ✇✐t❤♦✉t ❛tt❡♥t✐♦♥ t♦ ♦r❞❡r ♦❢ r❡❛❧✐s❛t✐♦♥ ✐❢ t❤✐♥❦ ♠♦r❡ r❡❝❡♥t ♦❜s❡r✈❛t✐♦♥s ❛r❡ ♠♦r❡ ✐♥❢♦r♠❛t✐✈❡✱ ♠❛② ❢✉❞❣❡✱ ✇❡✐❣❤t✐♥❣ t❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ❜② wt fiT ≡ ✶ T

t=✶ wt T

• t=✶

wtδ①t

✶ ❘♦❧❧✐♥❣ ✇✐♥❞♦✇ tr❡❛ts ❧❛st W ♦❜s❡r✈❛t✐♦♥s ❡q✉❛❧❧②✱ ❞✐s❝❛r❞✐♥❣ ❛❧❧ ❡❛r❧✐❡r

wt = ✶ ✐❢ t > (T − W ) wt = ✵ ✐❢ t ≤ (T − W )

✷ ❊①♣♦♥❡♥t✐❛❧ s♠♦♦t❤✐♥❣ ♣✐❝❦s ❛ ❞❡❝❛② ❢❛❝t♦r✱ λ ∈ [✵, ✶]✱ ❛♥❞ ✇❡✐❣❤ts ❜②

wt = (✶ − λ)T−t

❛♣♣r♦❛❝❤ ✉s❡❞ ❜② ❘✐s❦▼❡tr✐❝s ✭s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❑❛❧♠❛♥ ✜❧t❡r❄✮ ✇❤❡♥ T → ∞✱ ❝♦♥✈❡r❣❡s t♦ ❛ ●❆❘❈❍ ♠♦❞❡❧

✶✶✾ ✴ ✷✷✹

❊st✐♠❛t✐♥❣ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts

▲❡❝t✉r❡ ✻ ❡①❡r❝✐s❡s

▼❡✉❝❝✐ ❡①❡r❝✐s❡s

♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✻✳✷✳✶✱ ✻✳✷✳✹ ✭♥♦t ▼❆❚▲❆❇ ❝♦♠♣♦♥❡♥t✮✱ ✻✳✹✳✶✱ ✻✳✹✳✷✱ ✻✳✹✳✹ ▼❆❚▲❆❇✿ ✻✳✶ ✭❤✐♥t✿ ❙❴❇♦♥❞❘❡s✐❞✉❛❧▼♦❞❡❧ ✐s t❤❡ ❝♦rr❡❝t ✜❧❡♥❛♠❡✮✱ ✻✳✷✳✹ ✭▼❆❚▲❆❇ ❝♦♠♣♦♥❡♥t❀ ✇r✐t❡ ❛ ✇♦r❦❛r♦✉♥❞ ❢♦r ❝♦✈✷❝♦rr✭✮✱ ♦r ✉s❡ ❖❝t❛✈❡✬s✮✱ ✻✳✹✳✸✱ ✻✳✹✳✻✱ ✻✳✺

♣r♦❥❡❝t

❡①♣❡r✐♠❡♥t ✇✐t❤ ❞✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥✿ ❤♦✇ ✇❡❧❧ ❝❛♥ ②♦✉ ✜t t❤❡ ✸✲❞✐♠❡♥s✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ❢❡✇❡r ❞✐♠❡♥s✐♦♥s❄

✶✷✵ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s

❧❡t α ❜❡ ❛ ♣♦rt❢♦❧✐♦ ♦r ❛❧❧♦❝❛t✐♦♥✱ ❛♥ N✲✈❡❝t♦r ♦❢ ❛ss❡t ❤♦❧❞✐♥❣s✱ ❛♥❞ PT+τ,τ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ ♣r✐❝❡ ❞✐str✐❜✉t✐♦♥

✶ ✐♥✈❡st♦rs ❛r❡ ❝♦♥❝❡r♥❡❞ ❛❜♦✉t t❤❡✐r ♣♦rt❢♦❧✐♦✬s ♣❡r❢♦r♠❛♥❝❡ ❛t t❤❡

❤♦r✐③♦♥

❡✳❣✳ ❛❜s♦❧✉t❡ ✇❡❛❧t❤✱ r❡❧❛t✐✈❡ ✇❡❛❧t❤✱ ♥❡t ♣r♦✜ts ❝❛❧❧ t❤✐s t❤❡✐r ♦❜❥❡❝t✐✈❡✱ Ψα✱ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡

✷ ♥❡❡❞ t♦ ❝♦♥✈❡rt t❤✐s r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✐♥t♦ ❛ r❡❛❧ ♥✉♠❜❡r

❝❛❧❧ t❤✐s ❛♥ ✐♥❞❡① ♦❢ s❛t✐s❢❛❝t✐♦♥✱ S (α) ✭s✉♣♣r❡ss✐♥❣ ❞❡♣❡♥❞❡♥❝❡ ♦♥ Ψ✮

✶

❵❡❝♦♥♦♠✐st✬✿ ❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡①♣❡❝t❡❞ ✉t✐❧✐t②

✷

❵♣r❛❝t✐t✐♦♥❡rs✬✿ ❱❛❧✉❡ ❛t ❘✐s❦ ❜❛s❡❞ ♦♥ ❡✈❛❧✉❛t✐♥❣ q✉❛♥t✐❧❡s ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❛t ❣✐✈❡♥ ❝♦♥✜❞❡♥❝❡ ❧❡✈❡❧s

✸

❵✜♥❛♥❝❡✬✿ ❝♦❤❡r❡♥t ✐♥❞✐❝❡s✱ ❛♥❞ s♣❡❝tr❛❧ ✐♥❞✐❝❡s ❛s ❛ s✉❜s❡t✱ ✐♥❝❧✉❞✐♥❣ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✭❛❦❛ ❝♦♥❞✐t✐♦♥❛❧ ❱❛❧✉❡ ❛t ❘✐s❦✮

✶✷✶ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ■♥✈❡st♦rs✬ ♦❜❥❡❝t✐✈❡s

❚②♣✐❝❛❧ ♦❜❥❡❝t✐✈❡s✱ Ψα

✶ ❛❜s♦❧✉t❡ ✇❡❛❧t❤

Ψα = WT+τ (α) = α′PT+τ

❡✳❣✳ ✐♥✈❡st♦r ❝♦♥❝❡r♥❡❞ ❛❜♦✉t ❤❡r ✇❡❛❧t❤ ❛t r❡t✐r❡♠❡♥t

✷ r❡❧❛t✐✈❡ ✇❡❛❧t❤

Ψα = WT+τ (α) − γ (α) WT+τ (β) = α′❑PT+τ ✇❤❡r❡ γ (α) ≡ wT (α)

wT (β) ❛♥❞ ❑ ≡ ■N − ♣T β′ β′♣T

❡✳❣✳ ♠✉t✉❛❧ ❢✉♥❞ ♠❛♥❛❣❡r ❡✈❛❧✉❛t❡❞ ❛♥♥✉❛❧❧② ❛❣❛✐♥st ❛ ❜❡♥❝❤♠❛r❦

✸ ♥❡t ♣r♦✜ts

Ψα = WT+τ (α) − wT (α) = α′ (PT+τ − ♣T)

❡✳❣✳ tr❛❞❡r ❝♦♥❝❡r♥❡❞ ✇✐t❤ ❞❛✐❧② ♣r♦✜t ❛♥❞ ❧♦ss ✭P ✫ ▲✮❀ ♣r♦s♣❡❝t t❤❡♦r②

✶✷✸ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ■♥✈❡st♦rs✬ ♦❜❥❡❝t✐✈❡s

❇❡♥❝❤♠❛r❦✐♥❣✿ r❡❧❛t✐✈❡ ✇❡❛❧t❤ ♦❜❥❡❝t✐✈❡s

❣✐✈❡♥ ❛ r❡❧❛t✐✈❡ ✇❡❛❧t❤ ♦❜❥❡❝t✐✈❡✱ Ψα ≡ α′PT+τ − γβ′PT+τ ✇❤❡r❡ β ✐s ❛ ❜❡♥❝❤♠❛r❦ ♣♦rt❢♦❧✐♦ ❛♥❞ γ ≡ α′PT

β′PT ❡q✉❛❧✐s❡s ♣♦rt❢♦❧✐♦

❝♦sts ❡①♣❡❝t❡❞ ♦✈❡r♣❡r❢♦r♠❛♥❝❡ ✐s EOP (α) ≡ E {Ψα} tr❛❝❦✐♥❣ ❡rr♦r ✐s TE (α) ≡ Sd {Ψα} t❤❡ ✐♥❢♦r♠❛t✐♦♥ r❛t✐♦ ♥♦r♠❛❧✐s❡s ♦✉t♣❡r❢♦r♠❛♥❝❡ ❜② tr❛❝❦✐♥❣ ❡rr♦r✿ IR (α) ≡ EOP (α) TE (α) s❡❡ ❇❛❦❡r✱ ❇r❛❞❧❡②✱ ❛♥❞ ❲✉r❣❧❡r ✭✷✵✶✶✮ ❢♦r ❞❛♥❣❡rs ♦❢ ❜❡♥❝❤♠❛r❦✐♥❣ ✐♥ ❧♦♥❣✲♦♥❧② ♣♦rt❢♦❧✐♦s

✶✷✹ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ■♥✈❡st♦rs✬ ♦❜❥❡❝t✐✈❡s

❖❜❥❡❝t✐✈❡s✱ ✐♥ ❣❡♥❡r❛❧

✐♥ ❛❧❧ t❤❡ ♦❜❥❡❝t✐✈❡s ❝♦♥s✐❞❡r❡❞ Ψα = α′▼ ✇❤❡r❡ ▼ ≡ ❛ + ❇PT+τ ✐s t❤❡ ♠❛r❦❡t ✈❡❝t♦r✱ t❤❡ r❡❧❡✈❛♥t ❛✣♥❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ❤♦r✐③♦♥ ♣r✐❝❡s✱ ❛♥❞ ❇ ✐s ✐♥✈❡rt✐❜❧❡ ✇❤❛t ❛r❡ ❛ ❛♥❞ ❇ ❢♦r t❤❡ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡s❄ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ▼ ✐s ❡❛s✐❧② ❝♦♠♣✉t❡❞ ❢r♦♠ t❤❛t ♦❢ PT+τ φ▼ (ω) ≡E

• eiω′▼

= E

• eiω′(❛+❇PT+τ)

= E

• eiω′❛eiω′❇PT+τ
• =eiω′❛φPT+τ
• ❇′ω
• ❝❛♥ ❡❛s✐❧② s❤♦✇ t❤❛t Ψα ✐s

✶

❤♦♠♦❣❡♥❡♦✉s ♦❢ ❞❡❣r❡❡ ♦♥❡✿ Ψλα = λΨα

✷

❛❞❞✐t✐✈❡✿ Ψα+β = Ψα + Ψβ

❛s ♦❜❥❡❝t✐✈❡ ✐s ❛ r✈✱ ❤♦✇ ❝♦♠♣❛r❡ t✇♦ ♣♦rt❢♦❧✐♦s✱ α ❛♥❞ β❄

✶✷✺ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❙t♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡

❙t♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡

✶ ❛❧❧♦❝❛t✐♦♥ α str♦♥❣❧② ❞♦♠✐♥❛t❡s ❛❧❧♦❝❛t✐♦♥ β ✐✛

∀e ∈ E, Ψα > Ψβ

❛❧s♦ ❦♥♦✇♥ ❛s ③❡r♦ ♦r❞❡r ❞♦♠✐♥❛♥❝❡ ❤♦✇ ♦❢t❡♥ ❝❛♥ ✇❡ ❡①♣❡❝t t❤✐s❄

✷ ❛❧❧♦❝❛t✐♦♥ α ✇❡❛❦❧② ❞♦♠✐♥❛t❡s ❛❧❧♦❝❛t✐♦♥ β ✐✛

∀ψ ∈ (−∞, ∞) , FΨα (ψ) ≤ FΨβ (ψ) ⇔ QΨα (p) ≥ QΨβ (p) ∀p ∈ (✵, ✶)

❛❦❛ ✜rst ♦r❞❡r st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ✭❋❖❙❉✮ st✐❧❧ ✈❡r② r❛r❡

✸ ❛❧❧♦❝❛t✐♦♥ α s❡❝♦♥❞✲♦r❞❡r st♦❝❤❛st✐❝❛❧❧② ❞♦♠✐♥❛t❡s ❛❧❧♦❝❛t✐♦♥ β ✐✛

∀ψ ∈ (−∞, ∞) , ψ

−∞

(ψ − s) fΨα (s) ds ≥ ψ

−∞

(ψ − s) fΨβ (s) ds s♦ t❤❛t ❧♦✇❡r ♣❛rt✐❛❧ ❡①♣❡❝t❛t✐♦♥ ❢♦r ψα ❡①❝❡❡❞s t❤❛t ♦❢ ψβ ❢♦r ❛❧❧ ψ s❡❡ ▲❡✈② ✭✶✾✾✷✮ ❢♦r ❛ ❢✉❧❧ tr❡❛t♠❡♥t ♦❢ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡

✶✷✼ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ❞♦❡s ♥♦t ❣❡♥❡r❛t❡ ❝♦♠♣❧❡t❡ ♦r❞❡r ✇✐s❤✱ t❤❡r❡❢♦r❡✱ t♦ ❤❛✈❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ✐♥❞❡① ♦❢ s❛t✐s❢❛❝t✐♦♥ α → S (α) r✐s❦ ♠❡❛s✉r❡ ✐s −S❀ ♦♣❡r❛t✐♦♥❛❧✐s❡❞ ✈✐❛ r✐s❦ ❝❛♣✐t❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ✇❤❛t ❢❡❛t✉r❡s ✇♦✉❧❞ ❜❡ ❞❡s✐r❛❜❧❡ ❢♦r s✉❝❤ s✉♠♠❛r② st❛t✐st✐❝s t♦ ❤❛✈❡❄

✶

❢♦✉r ❛①✐♦♠s ❞❡✜♥❡ ❝♦❤❡r❡♥t ♠❡❛s✉r❡s ✭❆rt③♥❡r ❡t ❛❧✳✱ ✶✾✾✾✮

✷

t✇♦ ♠♦r❡ ❞❡✜♥❡ s♣❡❝tr❛❧ ♠❡❛s✉r❡s ✭❆❝❡r❜✐✱ ✷✵✵✷✮

✶✷✾ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❈♦❤❡r❡♥❝❡ ❛①✐♦♠ ✶✿ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡

✐❢ ❛❧❧♦❝❛t✐♦♥ ❜ ②✐❡❧❞s ❞❡t❡r♠✐♥✐st✐❝✱ ψ❜ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡ r❡q✉✐r❡s S (α + ❜) = S (α) + S (❜) = S (α) + ψ❜ t❤✐s✱ ✐♥ t✉r♥✱ ✐♠♣❧✐❡s

✶

❝♦♥st❛♥❝②✿ ♣❧✉❣ α = ✵ ✐♥t♦ t❤❡ ❛❜♦✈❡ ❢♦r S (❜) = ψ❜ ✭s❛t✐s❢❛❝t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❞❡t❡r♠✐♥✐st✐❝ ♦✉t❝♦♠❡ ✐s t❤❡ ♦✉t❝♦♠❡ ✐ts❡❧❢✮

✷

✐❢ ✉♥✐t ♦❢ ♠❡❛s✉r❡♠❡♥t ✐s ♠♦♥❡②✱ ♠♦♥❡②✲❡q✉✐✈❛❧❡♥❝❡✿ r❡❝❡✐✈✐♥❣ ❡①tr❛ ➾✶♠♥ ✐♥❝r❡❛s❡s s❛t✐s❢❛❝t✐♦♥ ✭r❡s♣ ❞❡❝r❡❛s❡s r✐s❦ ❝❛♣✐t❛❧✮ ❜② ➾✶♠♥

♥✳❜✳ ❛❞❞✐t✐✈❡ ♦❜❥❡❝t✐✈❡s ❞♦ ♥♦t ✐♠♣❧② ❛❞❞✐t✐✈❡ s❛t✐s❢❛❝t✐♦♥✿ Ψα+β = Ψα + Ψβ ⇒ S (α + β) = S (α) + S (β)

❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡ q✉❛♥t✐❧❡ ❝♦❤❡r❡♥t ✐♥❞✐❝❡s ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✶✸✵ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

▼♦♥❡②✲❡q✉✐✈❛❧❡♥❝❡ ✈ s❝❛❧❡✲✐♥✈❛r✐❛♥❝❡

❊①❛♠♣❧❡

✶ ❡①♣❡❝t❡❞ ✈❛❧✉❡✿ S (α) = E {Ψα} ✷ ❙❤❛r♣❡ r❛t✐♦✿ SR (α) = E{Ψα}

Sd{Ψα}

❲❤❡♥ ❤❛✈❡ ✇❡ s❡❡♥ ❛ ❙❤❛r♣❡ r❛t✐♦ ♣r❡✈✐♦✉s❧②❄ ✇❤✐❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❛r❡ ♠♦♥❡②✲❡q✉✐✈❛❧❡♥t❄ ❜② ❝♦♥tr❛st✱ ❞✐♠❡♥s✐♦♥❧❡ss s❝❛❧❡✲✐♥✈❛r✐❛♥❝❡ ✭❤♦♠♦❣❡♥❡✐t② ♦❢ ❞❡❣r❡❡ ③❡r♦✮ S (λα) = S (α) ∀λ > ✵ ♥♦r♠❛❧✐s❡s s✐③❡ ♦❢ ♣♦rt❢♦❧✐♦ ❛✇❛② ✇❤✐❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❛r❡ s❝❛❧❡✲✐♥✈❛r✐❛♥t❄

❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡ q✉❛♥t✐❧❡ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✶✸✶ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❈♦❤❡r❡♥❝❡ ❛①✐♦♠ ✷✿ s✉♣❡r✲❛❞❞✐t✐✈✐t②

❛♥ ✐♥❞❡① ♦❢ s❛t✐s❢❛❝t✐♦♥ ✐s s✉♣❡r✲❛❞❞✐t✐✈❡ ✐❢ t✇♦ ♣♦rt❢♦❧✐♦s ②✐❡❧❞ ❛ ❤✐❣❤❡r ✐♥❞❡① ♦❢ s❛t✐s❢❛❝t✐♦♥ t❤❛♥ t❤❡ ✐♥❞✐❝❡s ♦❢ t❤❡ ♣♦rt❢♦❧✐♦s ✐♥❞✐✈✐❞✉❛❧❧② S (α + β) ≥ S (α) + S (β) t❤✐s ✐s ❞❡s✐r❛❜❧❡ ❛s t❤❡ s✉♠♠❡❞ ♣♦rt❢♦❧✐♦ ✐s ❛t ❧❡❛st ❛s ❞✐✈❡rs✐✜❡❞ ❛s t❤❡ ✐♥❞✐✈✐❞✉❛❧ ♣♦rt❢♦❧✐♦s s✉♣❡r✲❛❞❞✐t✐✈❡ s❛t✐s❢❛❝t✐♦♥ ♠❡❛s✉r❡ ✐♠♣❧✐❡s ✇❤❛t s♦rt ♦❢ r✐s❦ ♠❡❛s✉r❡❄

❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡ q✉❛♥t✐❧❡ ❝♦❤❡r❡♥t ✐♥❞✐❝❡s ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧

❊①❛♠♣❧❡ ✭❊①♣❡❝t❡❞ ✈❛❧✉❡✮

S (α + β) ≡ E {Ψα+β} = E {Ψα} + E {Ψβ} = S (α) + S (β)

✶✸✷ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❈♦❤❡r❡♥❝❡ ❛①✐♦♠ ✸✿ ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t②

r❡s❝❛❧✐♥❣ ❛♥ ❛❧❧♦❝❛t✐♦♥ r❡s❝❛❧❡s t❤❡ ♦❜❥❡❝t✐✈❡ ✐♥ t❤❡ s❛♠❡ ✇❛② Ψλα = λΨα∀λ ≥ ✵ ✐❢ ❛♥ ✐♥❞❡① ♦❢ s❛t✐s❢❛❝t✐♦♥ r❡s❝❛❧❡s s✐♠✐❧❛r❧②✱ ✐t ✐s ❤♦♠♦❣❡♥❡♦✉s ✇✐t❤ ❞❡❣r❡❡ ♦♥❡ ♦r ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥♦✉s S (λα) = λS (α) ∀λ ≥ ✵ ❊✉❧❡r✬s ❤♦♠♦❣❡♥❡♦✉s ❢✉♥❝t✐♦♥ t❤❡♦r❡♠ ❛❧❧♦✇s s❛t✐s❢❛❝t✐♦♥ t♦ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ ❤♦ts♣♦ts✱ ❝♦♥tr✐❜✉t✐♦♥s ❢r♦♠ ❡❛❝❤ s❡❝✉r✐t② S (α) =

N

• n=✶

αn ∂S (α) ∂αn

❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡ q✉❛♥t✐❧❡ ❝♦❤❡r❡♥t ✐♥❞✐❝❡s ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✶✸✸ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

P♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t② + s✉♣❡r✲❛❞❞✐t✐✈✐t② ⇒ ❝♦♥❝❛✈✐t②

❛♥ ✐♥❞❡① ♦❢ s❛t✐s❢❛❝t✐♦♥ ✐s ❝♦♥❝❛✈❡ ✐✛ S (λα + (✶ − λ) β) ≥ λS (α) + (✶ − λ) S (β) ∀λ ∈ [✵, ✶] r❡❧❡✈❛♥❝❡✿ ❞✐✈❡rs✐✜❝❛t✐♦♥ ✈✐❛ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥s ♦❢ t✇♦ ♣♦rt❢♦❧✐♦s ✭❡✳❣✳ ❜✉❞❣❡t ❝♦♥str❛✐♥❡❞✮ ✐♥❝r❡❛s❡ s❛t✐s❢❛❝t✐♦♥ ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t② ❛♥❞ s✉♣❡r✲❛❞❞✐t✐✈✐t② ✐♠♣❧② ❝♦♥❝❛✈✐t② ✭r❡s♣✳ ❝♦♥✈❡①✐t② ❢♦r r✐s❦ ♠❡❛s✉r❡s✮ S (λα + (✶ − λ) β) ≥S (λα) + S ((✶ − λ) β) =λS (α) + (✶ − λ) S (β) ❜② s✉♣❡r✲❛❞❞✐t✐✈✐t②✱ ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t② r❡s♣❡❝t✐✈❡❧②

❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡ q✉❛♥t✐❧❡ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✶✸✹ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❈♦❤❡r❡♥❝❡ ❛①✐♦♠ ✹✿ ♠♦♥♦t♦♥✐❝✐t②

❜② ♥♦♥✲s❛t✐❛t✐♦♥✱ ❛ s❛t✐s❢❛❝t✐♦♥ ✐♥❞❡① s❛t✐s✜❡s ♠♦♥♦t♦♥✐❝✐t② ✐✛ Ψα ≥ Ψβ∀e ∈ E ⇒ S (α) ≥ S (β) t❤✉s✱ ♠♦♥♦t♦♥✐❝✐t② r❡q✉✐r❡s ❝♦♥s✐st❡♥❝② ✇✐t❤ str♦♥❣ ❞♦♠✐♥❛♥❝❡

α str♦♥❣❧② ❞♦♠✐♥❛t❡s β ⇒ S (α) ≥ S (β)

❛❣❛✐♥✱ s❡❡♠s ❛ s❡♥s✐❜❧❡ r❡q✉✐r❡♠❡♥t

❈♦✉♥t❡r❡①❛♠♣❧❡✿ ✷✵✵✻ ❙✇✐ss ❙♦❧✈❡♥❝② ❚❡st ✭❙❙❚✮

❛ ❢r❛♠❡✇♦r❦ ❢♦r ❞❡t❡r♠✐♥✐♥❣ ✏t❤❡ s♦❧✈❡♥❝② ❝❛♣✐t❛❧ r❡q✉✐r❡❞ ❢♦r ❛♥ ✐♥s✉r❛♥❝❡ ❝♦♠♣❛♥② ✳ ✳ ✳ ❚❤❡r❡ ❛r❡ s✐t✉❛t✐♦♥s ✇❤❡r❡ t❤❡ ❝♦♠♣❛♥② ✐s ❛❧❧♦✇❡❞ t♦ ❣✐✈❡ ❛✇❛② ❛ ♣r♦✜t❛❜❧❡ ♥♦♥✲r✐s❦② ♣❛rt ♦❢ ✐ts ❛ss❡t✲❧✐❛❜✐❧✐t② ♣♦rt❢♦❧✐♦ ✇❤✐❧❡ r❡❞✉❝✐♥❣ ✐ts t❛r❣❡t ❝❛♣✐t❛❧✳✑ ✭❋✐❧✐♣♦✈✐➣ ❛♥❞ ❱♦❣❡❧♣♦t❤✱ ✷✵✵✽✮ ❘❡❞✉❝❡s Ψα t♦ Ψβ ≤ Ψα✱ ❜✉t −S (β) ≤ −S (α)✳

❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡ q✉❛♥t✐❧❡ ❝♦❤❡r❡♥t ✐♥❞✐❝❡s ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✶✸✺ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❙♣❡❝tr❛❧ ❛①✐♦♠ ✺✿ ❧❛✇ ✐♥✈❛r✐❛♥❝❡

❧❛✇ ✐♥✈❛r✐❛♥t✿ S ❞❡♣❡♥❞s ♦♥❧② ♦♥ ❞✐str✐❜✉t✐♦♥ ♦❢ Ψα ✭❡✳❣✳ fΨα, FΨα, φΨα, QΨα✮ ❡q✉✐✈❛❧❡♥t t♦ ❡st✐♠❛❜❧❡ ❢r♦♠ ❡♠♣✐r✐❝❛❧ ❞❛t❛✿ ❜② ●❧✐✈❡♥❦♦✲❈❛♥t❡❧❧✐✱ ❛s s❛♠♣❧❡s ❜❡❝♦♠❡ ❧❛r❣❡✱ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ r✈s ②✐❡❧❞ t❤❡ s❛♠❡ S

❈♦✉♥t❡r❡①❛♠♣❧❡✿ ❣❡♥❡r❛❧ ❡q✉✐❧✐❜r✐✉♠ ♠❡❛s✉r❡s ♦❢ r✐s❦

✏t❤❡ r✐s❦ ♦❢ ❛ ♣♦rt❢♦❧✐♦ ❞❡♣❡♥❞s ♦♥ t❤❡ ♦t❤❡r ❛ss❡ts ♣r❡s❡♥t ✐♥ t❤❡ ❡❝♦♥♦♠② ✭t❤❡ ♠❛r❦❡t ♣♦rt❢♦❧✐♦✮ ✳ ✳ ✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠❡❛s✉r❡ ♦❢ r✐s❦ ♦❢ ❛ ♣♦rt❢♦❧✐♦ ✇♦✉❧❞ ❜❡ t❤❡ ❛♠♦✉♥t ♦❢ ❝❛s❤ ♥❡❡❞❡❞ t♦ s❡❧❧ t❤❡ r✐s❦ ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ ♣♦rt❢♦❧✐♦ t♦ t❤❡ ♠❛r❦❡t✑ ✭❈só❦❛✱ ❍❡r✐♥❣s✱ ❛♥❞ ❑ó❝③②✱ ✷✵✵✼✮ ▼❛r❦❡t ✐♠♣❛❝t✿ t❤✐♥ ♠❛r❦❡ts ✭✐❧❧✐q✉✐❞ ♦r ❡♠❡r❣✐♥❣✮ ♦r ❝♦rr❡❧❛t❡❞ ❜❡❤❛✈✐♦✉r ✭❙❤✐♥✱ ✷✵✶✵✮✱ ✐♥❝✳ ❛❧❣♦r✐t❤♠✐❝ ❝r♦✇❞✐♥❣ ❲♦rst ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ✭❆rt③♥❡r ❡t ❛❧✳✱ ✶✾✾✾✱ ❉❡✜♥✐t✐♦♥ ✺✳✷✮ ♥♦t ❧❛✇ ✐♥✈❛r✐❛♥t ❛s ❝♦♥❞✐t✐♦♥s ♦♥ st❛t❡ s♣❛❝❡ ✭❆❝❡r❜✐✱ ✷✵✵✷✮ ❍❛♥s♦♥✱ ❑❛s❤②❛♣✱ ❛♥❞ ❙t❡✐♥ ✭✷✵✶✶✮ ♦♥ ♠❛❝r♦♣r✉❞❡♥t✐❛❧ r❡❣✉❧❛t✐♦♥

❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡ q✉❛♥t✐❧❡ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✶✸✻ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❙♣❡❝tr❛❧ ❛①✐♦♠ ✻✿ ❝♦✲♠♦♥♦t♦♥✐❝ ❛❞❞✐t✐✈✐t②

❛❧❧♦❝❛t✐♦♥s α ❛♥❞ δ ❛r❡ ❝♦✲♠♦♥♦t♦♥✐❝ ✐❢ t❤❡✐r ♦❜❥❡❝t✐✈❡s ❛r❡ ❝♦✲♠♦♥♦t♦♥✐❝

❝♦✲♠♦♥♦t♦♥✐❝✐t②

❝♦♠❜✐♥✐♥❣ ❝♦✲♠♦♥♦t♦♥✐❝ ❛❧❧♦❝❛t✐♦♥s ❞♦❡s ♥♦t ♣r♦✈✐❞❡ ❣❡♥✉✐♥❡ ❞✐✈❡rs✐✜❝❛t✐♦♥ t❤✉s✱ ✐♥❞❡① ♦❢ s❛t✐s❢❛❝t✐♦♥ ✐s ❝♦✲♠♦♥♦t♦♥✐❝ ❛❞❞✐t✐✈❡ ✐✛ (α, δ) ❝♦✲♠♦♥♦t♦♥✐❝ ⇒ S (α + δ) = S (α) + S (δ) s✉❝❤ ✐♥❞✐❝❡s ❛r❡ ✏❞❡r✐✈❛t✐✈❡✲♣r♦♦❢✑

❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡ q✉❛♥t✐❧❡ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✶✸✼ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❧❛✇ ✐♥✈❛r✐❛♥t + ♠♦♥♦t♦♥✐❝ ⇒ ❝♦♥s✐st❡♥t ✇✐t❤ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡

❤❛✈❡ ❛♣♣❧✐❡❞ ♥♦♥✲s❛t✐❛t✐♦♥ t♦ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡✿ ❝❛♥ ❛❧s♦ ❛♣♣❧② t♦ ✇❡❛❦❡r ❝♦♥❝❡♣ts ♦❢ ❞♦♠✐♥❛♥❝❡ s♣❡❝tr❛❧ ♠❡❛s✉r❡ ⇒ ❝♦♥s✐st❡♥❝❡ ✇✐t❤ ✇❡❛❦ ✭♦r ✜rst ♦r❞❡r✮ ❞♦♠✐♥❛♥❝❡ QΨα (p) ≥ QΨβ (p) ∀p ∈ (✵, ✶) ⇒ S (α) ≥ S (β)

❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡ q✉❛♥t✐❧❡ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✶✸✽ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❉❡s✐❞❡r❛t✉♠✿ r✐s❦✲❛✈❡rs✐♦♥

❧❡t ❜ ❜❡ ❛♥ ❛❧❧♦❝❛t✐♦♥ ②✐❡❧❞✐♥❣ ❛ ❞❡t❡r♠✐♥✐st✐❝ ♦❜❥❡❝t✐✈❡✱ ψ❜ ❧❡t ❢ ❜❡ ❛ ❵❢❛✐r ❣❛♠❡✬ ❛❧❧♦❝❛t✐♦♥ ✇❤♦s❡ ♦❜❥❡❝t✐✈❡ ❤❛s E {Ψ❢ } = ✵ ❛♥ ✐♥❞❡① ♦❢ s❛t✐s❢❛❝t✐♦♥ ❞✐s♣❧❛②s r✐s❦✲❛✈❡rs✐♦♥ ✐✛ S (❜) ≥ S (❜ + ❢ ) t❤❡ r✐s❦✲♣r❡♠✐✉♠ ✐s t❤❡ ❞✐ss❛t✐s❢❛❝t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ r✐s❦② ❢ RP ≡ S (❜) − S (❜ + ❢ ) ✭✐❢ ♠♦♥❡②✲❡q✉✐✈❛❧❡♥t✱ ❤♦✇ ✐♥t❡r♣r❡t❄✮ ✐❢ S s❛t✐s✜❡s ❝♦♥st❛♥❝②✱ ❛♥❞ E {Ψα} ❡①✐sts✱ ❝❛♥ ❢❛❝t♦r ✐♥t♦ ❞❡t❡r♠✐♥✐st✐❝ ❛♥❞ ❵❢❛✐r ❣❛♠❡✬ ❝♦♠♣♦♥❡♥ts RP (α) ≡ E {Ψα} − S (α) ✭✇❤②❄✮ r✐s❦✲❛✈❡rs✐♦♥ ⇔ RP (α) ≥ ✵ ✭r❡❧❛t✐♦♥s❤✐♣ t♦ ❝♦♥❝❛✈✐t②❄✮

❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡ q✉❛♥t✐❧❡ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✶✸✾ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ▼❡❛s✉r❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

▲❡❝t✉r❡ ✼ ❡①❡r❝✐s❡s

▼❡✉❝❝✐ ❡①❡r❝✐s❡s

♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✼✳✷✳✶✱ ✼✳✷✳✷✱ ✼✳✸✳✷✱ ✼✳✸✳✸ ▼❆❚▲❆❇✿ ✼✳✶✳✶

♣r♦❥❡❝t✿ ❣✐✈❡♥ ②♦✉r ♣♦rt❢♦❧✐♦✱ ❝❛❧❝✉❧❛t❡✿

✶

✐ts ❛❜s♦❧✉t❡ ❡①♣❡❝t❡❞ ✇❡❛❧t❤

✷

✐ts ❙❤❛r♣❡ r❛t✐♦

✶✹✵ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥t ✭❡①♣❡❝t❡❞ ✉t✐❧✐t②✮

❊①♣❡❝t❡❞ ✉t✐❧✐t②

r❡❝❛❧❧✱ ❛ ♠❡❛s✉r❡ ♦❢ s❛t✐s❢❛❝t✐♦♥ ♠❛♣s ❢r♦♠ ❛♥ ❛❧❧♦❝❛t✐♦♥ t♦ ❛ ♥✉♠❜❡r✿ α → S (α) ✉t✐❧✐t② ❢✉♥❝t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡❛❝❤ r❡❛❧✐s❛t✐♦♥✱ ψ✱ s♦♠❡ ✉t✐❧✐t②✱ u (ψ) ❡①♣❡❝t❡❞ ✉t✐❧✐t② ✐s t❤❡r❡❢♦r❡ α → E {u (Ψα)} ≡

• R

u (ψ) fΨα (ψ) dψ ✭✇❤② ♥♦t ❥✉st ✉s❡ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡✱ E {Ψα}❄✮ ❛s ✉t✐❧✐t② ❤❛s ♥♦ ♠❡❛♥✐♥❣❢✉❧ ✉♥✐ts✱ ✐♥✈❡rt t♦ ♦❜t❛✐♥ ❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥t α → CE (α) ≡ u−✶ (E {u (Ψα)})

✶✹✷ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥t ✭❡①♣❡❝t❡❞ ✉t✐❧✐t②✮

Pr♦♣❡rt✐❡s ♦❢ ❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡

✶ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡❄ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡

u (ψ) = −e− ✶

ζ ψ ⇒ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ✇✇✇✳✺✳✸✮

✷ s✉♣❡r✲❛❞❞✐t✐✈✐t②❄ s✉♣❡r✲❛❞❞✐t✐✈✐t②

✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ♣✳✷✻✼✮✿ ♦♥❧② ❤♦❧❞s ❢♦r ❧✐♥❡❛r ✉t✐❧✐t②✱ u (ψ) ≡ ψ ✭✇❤❛t ❞♦ ❍❡♥♥❡ss② ❛♥❞ ▲❛♣❛♥ ✭✷✵✵✻✮ r❡s✉❧ts s❛②❄✮

✸ ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t②❄ ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t②

u (ψ) = ψ✶− ✶

γ , γ ≥ ✶ ⇒ ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t② ✭▼❡✉❝❝✐✱ ✷✵✵✺✱

✇✇✇✳✺✳✸✮

✹ ♠♦♥♦t♦♥✐❝✐t②❄ ♠♦♥♦t♦♥✐❝✐t②

✇❤❛t ❝♦♥❞✐t✐♦♥ ✐s r❡q✉✐r❡❞❄

✺ ❧❛✇✲✐♥✈❛r✐❛♥❝❡❄ ❧❛✇✲✐♥✈❛r✐❛♥❝❡ ✻ ❝♦✲♠♦♥♦t♦♥✐❝ ❛❞❞✐t✐✈✐t②❄ ❝♦✲♠♦♥♦t♦♥✐❝ ❛❞❞✐t✐✈✐t②

✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ♣✳✷✻✼✮✿ ♦♥❧② ❤♦❧❞s ❢♦r ❧✐♥❡❛r ✉t✐❧✐t②✱ u (ψ) ≡ ψ

✶✹✸ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥t ✭❡①♣❡❝t❡❞ ✉t✐❧✐t②✮

Pr♦♣❡rt✐❡s ♦❢ ❝❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥❝❡

✼ ❝♦♥❝❛✈✐t②❄ ❝♦♥❝❛✈✐t②

❛s s✉♠ ♦❢ ❝♦♥❝❛✈❡ ❢✉♥❝t✐♦♥s ✐s ❝♦♥❝❛✈❡✱ E {u (·)} ❝♦♥❝❛✈❡ ✐❢ u (·) ✐s ❜✉t t❤✐s ✐♠♣❧✐❡s t❤❛t u−✶ ✐s ❝♦♥✈❡①✱ s♦ u−✶ (E {u (·)}) ♥❡❡❞♥✬t ❜❡

✽ r✐s❦✲❛✈❡rs✐♦♥❄ r✐s❦✲❛✈❡rs✐♦♥

❛s CE s❛t✐s✜❡s ❝♦♥st❛♥❝② RP (α) ≡ E {Ψα} − CE (α) u (·) ❝♦♥❝❛✈❡ ⇔ RP (α) ≥ ✵ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ✇✇✇✳✺✳✸✮

✶✹✹ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈❡rt❛✐♥t②✲❡q✉✐✈❛❧❡♥t ✭❡①♣❡❝t❡❞ ✉t✐❧✐t②✮

❈♦♠♣✉t✐♥❣ CE (α) ≡ u−✶ (E {u (α′▼)})

❊①❛♠♣❧❡ ✭❊①♣♦♥❡♥t✐❛❧ ✉t✐❧✐t②❀ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ♠❛r❦❡ts✮

❡①♣♦♥❡♥t✐❛❧ ✉t✐❧✐t②✿ u (ψ) ≡ −e− ✶

ζ ψ ⇒ CE (α) = −ζ ❧♥

• φ▼
• i

ζ α

• ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ♠❛r❦❡ts✿

▼ ∼ N (µ, Σ) ⇒ CE (α) = α′µ − α′Σα

✷ζ

✉s✉❛❧❧② ♠✉st ❛♣♣r♦①✐♠❛t❡✱ ❡✳❣✳ s❡❝♦♥❞✲♦r❞❡r ❚❛②❧♦r s❡r✐❡s ❡①♣❛♥s✐♦♥ CE (α) ≡ E {Ψα} − RP (α) ≈ E {Ψα} − ✶ ✷A (E {Ψα}) Var {Ψα} ✇❤❡r❡ A (ψ) ≡ − u′′(ψ)

u′(ψ) ✐s t❤❡ ❆rr♦✇✲Pr❛tt ♠❡❛s✉r❡ ♦❢ ❛❜s♦❧✉t❡

r✐s❦✲❛✈❡rs✐♦♥

✶✹✺ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ◗✉❛♥t✐❧❡ ✭❱❛❧✉❡ ❛t ❘✐s❦✮

■♥tr♦❞✉❝t✐♦♥ t♦ ❱❛❘

❤♦✇ ♠✉❝❤ ❝❛♥ ✇❡ ❧♦s❡ ♦♥ ♦✉r tr❛❞✐♥❣ ♣♦rt❢♦❧✐♦ ❜② t♦♠♦rr♦✇✬s ❝❧♦s❡❄ ✭❆ttr✐❜✉t❡❞ t♦ ❉❡♥♥✐s ❲❡❛t❤❡rst♦♥❡✱ ♠♦t✐✈❛t✐♥❣ ❤✐s ❢❛♠♦✉s ✹✿✶✺ r❡♣♦rts ✭❆❧❧❡♥✱ ❇♦✉❞♦✉❦❤✱ ❛♥❞ ❙❛✉♥❞❡rs✱ ✷✵✵✹✮✮ ❣✐✈❡♥ ❛♥ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥✱ ❛♥❞ ❛ ❝♦♥✜❞❡♥❝❡ ❧❡✈❡❧✱ c✱ ❱❛❘ ✐s t❤❡ ♠❛①✐♠✉♠ ❧♦ss ♦✈❡r t❤❛t ♣❡r✐♦❞ c% ♦❢ t❤❡ t✐♠❡ ♣♦♣✉❧❛r✐t② ❣r❡✇ ❛❢t❡r ✶✾✾✻✱ ✇❤❡♥ ❏✳P✳ ▼♦r❣❛♥ ♣✉❜❧✐s❤❡❞ ✐ts ❱❛❘ ♠❡t❤♦❞♦❧♦❣② ✐♥ ✶✾✾✽✱ ❏✳P✳ ▼♦r❣❛♥ s♣✉♥ ♦✛ t❤❡ ❘✐s❦▼❡tr✐❝s ❣r♦✉♣ ♣r❡❢❡rr❡❞ ♠❡❛s✉r❡ ♦❢ ♠❛r❦❡t r✐s❦ ❛❞♦♣t❡❞ s✐♥❝❡ ❇❛s❡❧ ■■ ❝❛♥ ❝♦♥tr♦❧ ❜❛♥❦r✉♣t❝② r✐s❦ ✭❙❤✐♥✱ ✷✵✶✵✮ ❝r❡❞✐t r✐s❦ ✈❡rs✐♦♥ ❝❛❧❧❡❞ ♣♦t❡♥t✐❛❧ ❢✉t✉r❡ ❡①♣♦s✉r❡

✶✹✼ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ◗✉❛♥t✐❧❡ ✭❱❛❧✉❡ ❛t ❘✐s❦✮

❱❛❘ ✐❧❧✉str❛t❡❞

✶ − c VaRc (α) −Ψα ✭❧♦ss❡s✮ fΨα ✐❢ t❤❡ ♦❜❥❡❝t✐✈❡ ✐s ♥❡t ♣r♦✜ts Ψα ≡ WT+τ (α) − wT t❤❡♥✱ ❢r♦♠ ♦✉r ✈❡r❜❛❧ ❞❡✜♥✐t✐♦♥✱ P {−Ψα ≥ VaRc (α)} = ✶ − c P {Ψα ≤ −VaRc (α)} = ✶ − c FΨα (−VaRc (α)) = ✶ − c − VaRc (α) = QΨα (✶ − c) VaRc (α) ≡ −QΨα (✶ − c)

q✉❛♥t✐❧❡

❛♣♣❧✐❡s ❡q✉❛❧❧② t♦ ❛♥② ♦t❤❡r Ψα Qc (α) ≡ QΨα (✶ − c) ✭✇❤② ♥♦ ♥❡❣❛t✐✈❡ s✐❣♥❄✮

✶✹✽ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ◗✉❛♥t✐❧❡ ✭❱❛❧✉❡ ❛t ❘✐s❦✮

Pr♦♣❡rt✐❡s ♦❢ q✉❛♥t✐❧❡ ♠❡❛s✉r❡s

✶ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡❄ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡

✐♥t✉✐t✐♦♥❄ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ✇✇✇✳✺✳✹✮

✷ s✉♣❡r✲❛❞❞✐t✐✈✐t②❄ s✉♣❡r✲❛❞❞✐t✐✈✐t②

❢✉❧❧② ❝♦♥❝❡♥tr❛t❡❞ ♣♦rt❢♦❧✐♦s ❝❛♥ ❤❛✈❡ ❧♦✇❡r ❱❛❘ t❤❛♥ ❢✉❧❧② ❞✐✈❡rs✐✜❡❞ ♦♥❡s ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ❊①❛♠♣❧❡ ✷✳✷✺✮

❱❛❘ ❢❛✐❧

t❤✐s ❢❛✐❧✉r❡ ♣r♦♠♣t❡❞ s❡❛r❝❤ ❢♦r ❛❧t❡r♥❛t✐✈❡s ❜✉t ❤♦❧❞s ❢♦r ❡❧❧✐♣t✐❝❛❧ ♠❛r❦❡ts ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ❚❤❡♦r❡♠ ✽✳✷✽✭✷✮✮ ❊♠❜r❡❝❤ts✱ ▲❛♠❜r✐❣❣❡r✱ ❛♥❞ ❲üt❤r✐❝❤ ✭✷✵✵✾✮ ❢♦r ❞❡t❛✐❧❡❞ ❞✐s❝✉ss✐♦♥ ♦❢ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ s✉♣❡r✲❛❞❞✐t✐✈✐t② ❢❛✐❧✉r❡s ❢♦r ❱❛❘

❡①♣❡❝t❡❞ ✈❛❧✉❡ ✸ ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t②❄ ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t②

✐♥t✉✐t✐♦♥❄ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ✇✇✇✳✺✳✹✮ ∴ ❊✉❧❡r ❝♦♥❞✐t✐♦♥ ❤♦❧❞s

✹ ♠♦♥♦t♦♥✐❝✐t②❄ ♠♦♥♦t♦♥✐❝✐t② ✺ ❧❛✇✲✐♥✈❛r✐❛♥❝❡❄ ❧❛✇✲✐♥✈❛r✐❛♥❝❡ ✻ ❝♦✲♠♦♥♦t♦♥✐❝ ❛❞❞✐t✐✈✐t②❄ ❝♦✲♠♦♥♦t♦♥✐❝ ❛❞❞✐t✐✈✐t②

✐♥t✉✐t✐♦♥❄ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ✇✇✇✳✺✳✹✮ t❤✉s✱ ❝♦♥s✐st❡♥t ✇✐t❤ ✜rst ♦r❞❡r st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ✭❝♦✉♥t❡r✲❡①❛♠♣❧❡s ❢♦r s❡❝♦♥❞ ❛♥❞ ❤✐❣❤❡r ♦r❞❡rs ▼❡✉❝❝✐ ✭✷✵✵✺✱ ♣✳✷✼✾✮✮

✶✹✾ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ◗✉❛♥t✐❧❡ ✭❱❛❧✉❡ ❛t ❘✐s❦✮

Pr♦♣❡rt✐❡s ♦❢ q✉❛♥t✐❧❡ ♠❡❛s✉r❡s

✼ ❝♦♥❝❛✈✐t②❄ ❝♦♥❝❛✈✐t②

❢❛✐❧✉r❡ r❡❧❛t❡❞ t♦ t❤❛t ♦❢ s✉♣❡r✲❛❞❞✐t✐✈✐t②✱ ❛❜♦✈❡❄

✽ r✐s❦✲❛✈❡rs✐♦♥❄ r✐s❦✲❛✈❡rs✐♦♥

RP (α) ❝❛♥ t❛❦❡ ♦♥ ❛♥② s✐❣♥

✶✺✵ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ◗✉❛♥t✐❧❡ ✭❱❛❧✉❡ ❛t ❘✐s❦✮

❈♦♠♣✉t✐♥❣ Qc (α) ≡ Qα′▼ (✶ − c)

❊①❛♠♣❧❡ ✭◆❡t ♣r♦✜ts ❛♥❞ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ♠❛r❦❡ts✮

PT+τ ∼ N (µ, Σ) ❛♥❞ Ψα ≡ α′▼ ⇒ Ψα ∼ N

• µα, σ✷

α

• Qc (α) = µα +

√ ✷σα ❡r❢−✶ (✶ − ✷c) ✉s✉❛❧❧② ♠✉st ❛♣♣r♦①✐♠❛t❡

✶

❞❡❧t❛✲❣❛♠♠❛ ❛♣♣r♦①✐♠❛t✐♦♥✿ s❡❝♦♥❞ ♦r❞❡r ❚❛②❧♦r s❡r✐❡s ❡①♣❛♥s✐♦♥

✷

❈♦r♥✐s❤✲❋✐s❤❡r ❡①♣❛♥s✐♦♥✿ ❡①♣❛♥s✐♦♥ ✇❤♦s❡ t❡r♠s ❛r❡ t❤❡ r✈✬s ♠♦♠❡♥ts

✸

❡①tr❡♠❡ ✈❛❧✉❡ t❤❡♦r② ❛s c → ✶✿ ❥✉st ✜t t❤❡ t❛✐❧ ✭❡✳❣✳ ✉s✐♥❣ ❛ ❣❡♥❡r❛❧✐s❡❞ P❛r❡t♦ ❞✐str✐❜✉t✐♦♥✮

s✐♠✉❧❛t❡❞ ❞❛t❛✿ s♦rt ❜② Ψα ❛♥❞ ♣✐❝❦ s❝❡♥❛r✐♦ ♥❡❛r❡st ❞❡s✐r❡❞ q✉❛♥t✐❧❡

• ♦✉r✐❡r✱ ❋❛r❦❛s✱ ❛♥❞ ❆❜❜❛t❡ ✭✷✵✵✾✮ ❛♣♣❧✐❡s ❱❛❘ t♦ ■t❛❧✐❛♥ ❜❛♥❦ ❞❛t❛❀

s❡❡ ❑r✐t③♠❛♥ ✭✷✵✶✶✮ ♦♥ t❤♦✉❣❤t❢✉❧ ✈✳ ♥❛ï✈❡ ✉s❡

✶✺✶ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦❤❡r❡♥t ✐♥❞✐❝❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❙♣❡❝tr❛❧ ✐♥❞✐❝❡s ✭❆❝❡r❜✐✱ ✷✵✵✷✮

❡①✐st✐♥❣ ✐♥❞✐❝❡s ❡✐t❤❡r s❛t✐s✜❡❞ ♦r ❢❛✐❧❡❞ t♦ s❛t✐s❢② ❝❡rt❛✐♥ ♣r♦♣❡rt✐❡s ❜♦t❤ ❡①♣❡❝t❡❞ ✉t✐❧✐t② ✭✐♥ ❣❡♥❡r❛❧✮ ❛♥❞ q✉❛♥t✐❧❡ ♠❡❛s✉r❡s ❢❛✐❧ t♦ s❛t✐s❢② s✉♣❡r✲❛❞❞✐t✐✈✐t②✱ ❝♦♥❝❛✈✐t② ❜♦t❤ ♠❛② t❤❡r❡❢♦r❡ ❢❛✐❧ t♦ ✉♥❞❡rst❛♥❞ ♠♦t✐✈❡s ❢♦r ❞✐✈❡rs✐✜❝❛t✐♦♥ ❝♦❤❡r❡♥t ✐♥❞✐❝❡s ❞❡s✐❣♥❡❞ t♦ s❛t✐s❢② t❤❡s❡ ♣r♦♣❡rt✐❡s ❣✐✈❡♥ ❛ ❝♦❤❡r❡♥t ✐♥❞❡①✱ ❤♦✇ ❝❛♥ ♦t❤❡rs ❜❡ ❣❡♥❡r❛t❡❞❄ q✉❡st✐♦♥ ❣❛✈❡ r✐s❡ t♦ s♣❡❝tr❛❧ ✐♥❞✐❝❡s✱ ❛ s✉❜❝❧❛ss ♦❢ ❝♦❤❡r❡♥t ✐♥❞✐❝❡s

✐♥ s❛t✐s❢②✐♥❣ ❛❞❞✐t✐♦♥❛❧ t✇♦ ❛①✐♦♠s✱ ❛❧s♦ s❛t✐s❢② r✐s❦✲❛✈❡rs✐♦♥

✶✺✸ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦❤❡r❡♥t ✐♥❞✐❝❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❊①♣❡❝t❡❞ ✈❛❧✉❡ ❛s ❛ s♣❡❝tr❛❧ ♠❡❛s✉r❡ ♦❢ s❛t✐s❢❛❝t✐♦♥

❚❤❡♦r❡♠

❚❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡✱ E {Ψα}✱ ✐s ❛ s♣❡❝tr❛❧ ♠❡❛s✉r❡ ♦❢ s❛t✐s❢❛❝t✐♦♥✳

Pr♦♦❢✳

✶ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡✿

E {Ψα + ψ❜} = E {Ψα} + E {ψ❜} = E {Ψα} + ψ❜

✷ s✉♣❡r✲❛❞❞✐t✐✈✐t②✿ E {Ψα+β} = E {Ψα} + E {Ψβ} ✸ ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t②✿ E {Ψλα} = E {λΨα} = λE {Ψα} ✹ ♠♦♥♦t♦♥✐❝✐t②✿ Ψα ≥ Ψβ∀e ∈ E ⇒ E {Ψα} ≥ E {Ψβ} ✺ ❧❛✇ ✐♥✈❛r✐❛♥❝❡✿ E {Ψα} ≡

• R ψfΨα (ψ) dψ

✻ ❝♦✲♠♦♥♦t♦♥✐❝ ❛❞❞✐t✐✈✐t②✿ ❛❞❞✐t✐✈❡ ❢♦r ❛♥② α, β✱ ♥♦t ❥✉st ❝♦✲♠♦♥♦t♦♥✐❝ ✶✺✹ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦❤❡r❡♥t ✐♥❞✐❝❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❊①♣❡❝t❡❞ ✈❛❧✉❡ ❛s ❛♥ ❛✈❡r❛❣❡ ♦❢ q✉❛♥t✐❧❡s

▲❡♠♠❛

E {Ψα} ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s t❤❡ ✉♥✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♦❢ t❤❡ q✉❛♥t✐❧❡s E {Ψα} ≡

• R

ψfΨα (ψ) dψ = ✶

✵

QΨα (p) dp

♣r♦♦❢

r❡❝❛❧❧✿ t❤❡ q✉❛♥t✐❧❡ ✐ts❡❧❢ ✐s ♥♦t s✉♣❡r✕❛❞❞✐t✐✈❡

q✉❛♥t✐❧❡

❜✉t ❡①♣❡❝t❡❞ ✈❛❧✉❡✱ ❛s t❤❡ ❛✈❡r❛❣❡ ♦❢ ❛❧❧ q✉❛♥t✐❧❡s✱ ✐s

✇❤❛t ❛❜♦✉t ❛♥ ❛✈❡r❛❣❡ ♦✈❡r t❤❡ ✇♦rst s❝❡♥❛r✐♦s❄

✶✺✺ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦❤❡r❡♥t ✐♥❞✐❝❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❊①♣❡❝t❡❞ s❤♦rt❢❛❧❧

❡①♣❡❝t❡❞ ✈❛❧✉❡ ❛✈❡r❛❣❡s ♦✈❡r ❛❧❧ s❝❡♥❛r✐♦s E {Ψα} ≡ ✶

✵

QΨα (p) dp ♥♦✇ ❞❡✜♥❡ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ t♦ ❛✈❡r❛❣❡ ♦✈❡r t❤❡ ✇♦rst s❝❡♥❛r✐♦s✱ ESc {α} ≡ ✶ ✶ − c ✶−c

✵

QΨα (p) dp = E {Ψα |Ψα ≤ Qc (α)} ✇❤❡r❡ c ∈ [✵, ✶] ✐♥❞❡①❡s t❤❡ ❝♦♥✜❞❡♥❝❡ ❧❡✈❡❧ s♦✉❣❤t ✇❤② ✐s t❤❡ (✶ − c)−✶ t❡r♠ ♣r❡s❡♥t❄ ✇❤❡♥ fΨα ✐s s♠♦♦t❤ ✭❆❝❡r❜✐ ❛♥❞ ❚❛s❝❤❡✱ ✷✵✵✷✮✱ ❡q✉✐✈❛❧❡♥t t♦

t❛✐❧ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ✭❚❈❊✮ ❝♦♥❞✐t✐♦♥❛❧ ✈❛❧✉❡ ❛t r✐s❦ ✭❈❱❛❘✮

✶✺✻ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦❤❡r❡♥t ✐♥❞✐❝❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

Pr♦♣❡rt✐❡s ♦❢ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧

✶ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡❄ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡

❢r♦♠ ❧✐♥❡❛r✐t② ♦❢ ✐♥t❡❣r❛❧ ❛♥❞ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡ ♦❢ q✉❛♥t✐❧❡

✷ s✉♣❡r✲❛❞❞✐t✐✈✐t②❄ s✉♣❡r✲❛❞❞✐t✐✈✐t②

❛s ❛✈❡r❛❣✐♥❣ ♦✈❡r t❛✐❧✱ ❝❛♥✬t ❵❜♦♠❜✬ ✐t ❛s ❝❛♥ ❱❛❘ s❡❡ ❆❝❡r❜✐ ❛♥❞ ❚❛s❝❤❡ ✭✷✵✵✷✱ Pr♦♣♦s✐t✐♦♥ ❆✳✶✮

✸ ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t②❄ ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t②

✐♥t❡❣r❛❧ ✐s ❧✐♥❡❛r❀ q✉❛♥t✐❧❡ ✐s ♣♦s✐t✐✈❡❧② ❤♦♠♦❣❡♥❡♦✉s❀ ❛❣❛✐♥✱ ❊✉❧❡r ❝♦♥❞✐t✐♦♥ ❤♦❧❞s

✹ ♠♦♥♦t♦♥✐❝✐t②❄ ♠♦♥♦t♦♥✐❝✐t② ✺ ❧❛✇✲✐♥✈❛r✐❛♥❝❡❄ ❧❛✇✲✐♥✈❛r✐❛♥❝❡ ✻ ❝♦✲♠♦♥♦t♦♥✐❝ ❛❞❞✐t✐✈✐t②❄ ❝♦✲♠♦♥♦t♦♥✐❝ ❛❞❞✐t✐✈✐t②

❢r♦♠ ❧✐♥❡❛r✐t② ♦❢ ✐♥t❡❣r❛❧ ❛♥❞ ❝♦✲♠♦♥♦t♦♥✐❝ ❛❞❞✐t✐✈✐t② ♦❢ q✉❛♥t✐❧❡

✶✺✼ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦❤❡r❡♥t ✐♥❞✐❝❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

Pr♦♣❡rt✐❡s ♦❢ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧

✼ ❝♦♥❝❛✈✐t②❄ ❝♦♥❝❛✈✐t②

❢r♦♠ ♣♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t② ❛♥❞ s✉♣❡r✕❛❞❞✐t✐✈✐t②

✽ r✐s❦✲❛✈❡rs✐♦♥❄ r✐s❦✲❛✈❡rs✐♦♥

❢r♦♠ t❤❡ ♦t❤❡r ♣r♦♣❡rt✐❡s ♦❢ s♣❡❝tr❛❧ ✐♥❞✐❝❡s

t❤✉s ESc (α) ✐s ❛ s♣❡❝tr❛❧ ♠❡❛s✉r❡ ♦❢ s❛t✐s❢❛❝t✐♦♥ ❢♦r ❛♥② c ∈ [✵, ✶]

✶✺✽ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦❤❡r❡♥t ✐♥❞✐❝❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❇✉✐❧❞✐♥❣ s♣❡❝tr❛❧ ✐♥❞✐❝❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

t♦ ❣❡♥❡r❛t❡ ❢❛♠✐❧② ♦❢ s♣❡❝tr❛❧ ✐♥❞✐❝❡s✱ ❜❡❣✐♥ ✇✐t❤ s♣❡❝tr❛❧ ❜❛s✐s ✉s❡ ESc (α) t♦ ❣❡♥❡r❛t❡ ❝❧❛ss ✭❆❝❡r❜✐ ✭✷✵✵✷✮✱ ▼❡✉❝❝✐ ✭✷✵✵✺✱ ✇✇✇✳✺✳✺✮✮ Spcϕ (α) ≡ ✶

✵

ϕ (p) QΨα (p) dp ✇❤❡r❡ t❤❡ s♣❡❝tr✉♠✱ ϕ✱ ✭✇❡❛❦❧②✮ ❞❡❝r❡❛s❡s t♦ ϕ (✶) = ✵✱ ❛♥❞ s❡ts ✶

✵ ϕ (p) dp = ✶

ϕ ❣✐✈❡s ♠♦r❡ ✇❡✐❣❤t t♦ t❤❡ ❧♦✇❡st q✉❛♥t✐❧❡s ✭t❤❡ ✇♦rst ♦✉t❝♦♠❡s✮

❛♥② s♣❡❝tr❛❧ ✐♥❞❡① ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② ❛ ϕ s❛t✐s❢②✐♥❣ t❤❡ ❛❜♦✈❡

❊①❛♠♣❧❡

ϕESc (p) ≡

✶ ✶−c H(c−✶) (−p)✱ ✇❤❡r❡ H(x) ✐s t❤❡ ❍❡❛✈✐s✐❞❡ st❡♣ ❢✉♥❝t✐♦♥✳

❞r❛✇ ϕ ❢♦r ❡①♣❡❝t❡❞ ✈❛❧✉❡ ❝❛♥ ②♦✉ ❞r❛✇ ϕ ❢♦r Qc (α)❄ ❲❤② ♦r ✇❤② ♥♦t❄ ❢♦r ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s✱ q✳✈✳ ❊❧❧✐s♦♥ ❛♥❞ ❙❛r❣❡♥t ✭✷✵✶✷✮

✶✺✾ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦❤❡r❡♥t ✐♥❞✐❝❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

❈♦♠♣✉t✐♥❣ Spcϕ (α) ≡ ✶

✵ ϕ (p) Qα′▼ (p) dp

❊①❛♠♣❧❡ ✭◆❡t ♣r♦✜ts ❛♥❞ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ♠❛r❦❡ts✮

PT+τ ∼ N (µ, Σ) ❛♥❞ Ψα ≡ α′▼ ⇒ Ψα ∼ N

• µα, σ✷

α

• Qα′▼ (α) = µα +

√ ✷σα ❡r❢−✶ (✶ − ✷c) ⇒ Spcα = µα + √ ✷σα ✶

✵

ϕ (p) ❡r❢−✶ (✷p − ✶) dp ✉s✉❛❧❧② ❛♣♣r♦①✐♠❛t❡✱ ❡✳❣✳ ❞❡❧t❛✲❣❛♠♠❛ ❛♣♣r♦①✐♠❛t✐♦♥ ♦r ❈♦r♥✐s❤✲❋✐s❤❡r ❡①♣❛♥s✐♦♥ ❢♦r ESc (α)✱ ❝❛♥ ❛❧s♦ ✉s❡ ❡①tr❡♠❡ ✈❛❧✉❡ t❤❡♦r② ❛s c → ✶ s✐♠✉❧❛t❡❞ ❞❛t❛✿ s♦rt ❜② Ψα ❛♥❞ ❛✈❡r❛❣❡ s❝❡♥❛r✐♦s ❜❡❧♦✇ Qc (α)

✶✻✵ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦❤❡r❡♥t ✐♥❞✐❝❡s ♦❢ s❛t✐s❢❛❝t✐♦♥

▲❡❝t✉r❡ ✽ ❡①❡r❝✐s❡s

▼❡✉❝❝✐ ❡①❡r❝✐s❡s

♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✼✳✹✳✶✱ ✼✳✺✳✶ ✭♥♦t ▼❆❚▲❆❇✮ ▼❆❚▲❆❇✿ ✼✳✹✳✶✱ ✼✳✹✳✷✱ ✼✳✹✳✸✱ ✼✳✺✳✶ ✭▼❆❚▲❆❇✮✱ ✼✳✺✳✷

♣r♦❥❡❝t✿ ❣✐✈❡♥ ②♦✉r ♣♦rt❢♦❧✐♦ ❛♥❞ ❛ ✇❡❡❦❧② ❤♦r✐③♦♥✱ ❝❛❧❝✉❧❛t❡ t❤❡ ✺✪✱ ✶✪ ❛♥❞ ✵✳✶✪ ❱❛❘s

✶✻✶ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ■♥tr♦❞✉❝t✐♦♥

❚❤❡ ✐♥❣r❡❞✐❡♥ts

✶ ❝♦❧❧❡❝t✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ ✐♥✈❡st♦r✬s ♣r♦✜❧❡✱ P ✶

❡①✐st✐♥❣ ♣♦rt❢♦❧✐♦✱ α(✵)

✷

✐♥✈❡st♠❡♥t ❤♦r✐③♦♥✱ T + τ

✸

♠❛r❦❡ts ♦❢ ✐♥t❡r❡st ✭❡✳❣✳ ❛❧t❡r♥❛t✐✈❡s✱ ♠✉t✉❛❧ ❢✉♥❞s✱ ❡t❝✳✮

✹

♦❜❥❡❝t✐✈❡✱ Ψα

✺

r✐s❦✴s❛t✐s❢❛❝t✐♦♥ ✐♥❞❡①✱ S (α)

✷ ❝♦❧❧❡❝t✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ ♠❛r❦❡t✱ iT ✶

❝✉rr❡♥t s❡❝✉r✐t✐❡s ♣r✐❝❡s✱ ♣T

✷

❤♦r✐③♦♥ s❡❝✉r✐t✐❡s ♣r✐❝❡s✱ PT+τ ✭❤♦✇❄✮

✸

tr❛♥s❛❝t✐♦♥ ❝♦sts✱ T

• α(✵), α
• ✶✻✸ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ■♥tr♦❞✉❝t✐♦♥

❖♣t✐♠❛❧ ❛❧❧♦❝❛t✐♦♥s

❛♥ ❛❧❧♦❝❛t✐♦♥ ✐s t❤❡r❡❢♦r❡ α : [P, iT] → RN ❛♥ ♦♣t✐♠❛❧ ❛❧❧♦❝❛t✐♦♥ ✐s α∗ ≡ ❛r❣♠❛① S (α) s✳t✳ α ∈ C ✇❤❡r❡ C ❞❡✜♥❡s t❤❡ ❝♦♥tr❛✐♥t s❡t

♣′

Tα + T

• α(✵), α
• − b ≤ ✵ ✇❤❡r❡ b ✐s ❛ ❜✉❞❣❡t ❝♦♥str❛✐♥t

s❡❝♦♥❞❛r② ♦❜❥❡❝t✐✈❡s ✭❡✳❣✳ ❱❛❘ t❛r❣❡ts✮

t❤❡s❡ ❛r❡ ♥♦t ❣❡♥❡r❛❧❧② ♣♦ss✐❜❧❡ t♦ s♦❧✈❡ ❛♥❛❧②t✐❝❛❧❧②

✶✻✹ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦♥str❛✐♥❡❞ ♦♣t✐♠✐s❛t✐♦♥

❈♦♥str❛✐♥❡❞ ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠s

t❤❡ ❣❡♥❡r❛❧ ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠ ③∗ ≡ ❛r❣♠✐♥ Q (③) s✳t✳ ③ ∈ Rn, fi (③) ≤ ✵, ❢♦r i = ✶, . . . , m ✇❤❡r❡

Q (③) ✐s ❛♥ ❛r❜✐tr❛r② ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ t❤❡ fi (③) ❛r❡ ❛r❜✐tr❛r② ❝♦♥str❛✐♥ts ③ ❛r❡ t❤❡ ❝❤♦✐❝❡ ✈❛r✐❛❜❧❡s

✐s ❛ ❣❧♦❜❛❧ ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠✱ ❤❡♥❝❡ NP✲❤❛r❞ ❝♦♥✈❡① ♣r♦❣r❛♠♠✐♥❣ ✐s ❛ s✉❜s❡t s✉❝❤ t❤❛t

Q (③) ✐s ❛ ❝♦♥✈❡① ❢✉♥❝t✐♦♥ t❤❡ fi (③) ❛r❡ ❛❧s♦ ❝♦♥✈❡①

❝♦♥✈❡① ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠s ❝❛♥ ❜❡ ❡✣❝✐❡♥t❧② s♦❧✈❡❞ ✭❡✳❣✳ ✐♥ P✮✱ ❤❛✈❡ ❦♥♦✇♥ ✉♥✐q✉❡♥❡ss ❇♦②❞ ❛♥❞ ❱❛♥❞❡♥❜❡r❣❤❡ ✭✷✵✵✹✮ ✐s ❛ st❛♥❞❛r❞✱ ✇❡❧❧ s✉♣♣♦rt❡❞ t❡①t ✭q✳✈✳ t❤❡ ♦♣❡♥ ❝♦✉rs❡s✱ ❤❡r❡ ❛♥❞ ❤❡r❡✮

✶✻✻ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦♥str❛✐♥❡❞ ♦♣t✐♠✐s❛t✐♦♥

❈♦♥❡ ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠s

❝♦♥❡ ♣r♦❣r❛♠♠✐♥❣ ✐s ❛ s✉❜s❡t ♦❢ ❝♦♥✈❡① ♣r♦❣r❛♠♠✐♥❣ s✉❝❤ t❤❛t

Q (③) ✐s ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥✱ ❝′③ t❤❡ fi (③) ❞❡✜♥❡ ❛ ❝♦♥❡✱ K

❉❡✜♥✐t✐♦♥ ✭❈♦♥❡✮

✶ ❝❧♦s❡❞ ✉♥❞❡r ♣♦s✐t✐✈❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥✿ ② ∈ K, λ ≥ ✵ ⇒ λ② ∈ K ✷ ❝❧♦s❡❞ ✉♥❞❡r ❛❞❞✐t✐♦♥✿ ①, ② ∈ K ⇒ ① + ② ∈ K ✸ ❵♣♦✐♥t❡❞✬✿ ② ∈ K ⇒ −② ∈ K

s♦❢t✇❛r❡ ♣❛❝❦❛❣❡s ✉s✐♥❣ ✐♥t❡r✐♦r✲♣♦✐♥t ♠❡t❤♦❞s ✭❇♦②❞ ❛♥❞ ❱❛♥❞❡♥❜❡r❣❤❡✱ ✷✵✵✹✱ ❝❤✳✶✶✮ ❝❛♥ ❡✣❝✐❡♥t❧② s♦❧✈❡ t❤❡s❡ ✐♥❝❧✉❞❡s ✇❡❧❧✲❦♥♦✇♥ ❝❧❛ss❡s ❛s s♣❡❝✐❛❧ ❝❛s❡s

✶✻✼ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦♥str❛✐♥❡❞ ♦♣t✐♠✐s❛t✐♦♥

❈♦♥❡ ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠s

✶ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣

s❡ts ❇③ − ❜ ≥ ✵ s♦ t❤❛t K ≡ RM

+ ✱ t❤❡ ♣♦s✐t✐✈❡ ♦rt❤❛♥t

s✐♠♣❧❡①✱ ✐♥t❡r✐♦r ♣♦✐♥t ♠❡t❤♦❞s ❜♦t❤ ♣❡r❢♦r♠ ✇❡❧❧

✷ q✉❛❞r❛t✐❝❛❧❧② ❝♦♥str❛✐♥❡❞ q✉❛❞r❛t✐❝ ♣r♦❣r❛♠♠✐♥❣ ✭◗❈◗P✮ ✐♥❝❧✉❞❡s ▲P

❤❛s q✉❛❞r❛t✐❝ ♦❜❥❡❝t✐✈❡ Q = ③′❙(✵)③ + ✷✉′

(✵)③ + v(✵)

❜✉t ❝❛♥ ✐♥tr♦❞✉❝❡ ❛✉①✐❧✐❛r② ✈❛r✐❛❜❧❡ t♦ tr❛♥s❢♦r♠ Q ✐♥t♦ ❧✐♥❡❛r

✸ s❡❝♦♥❞✲♦r❞❡r ❝♦♥❡ ♣r♦❣r❛♠♠✐♥❣ ✭❙❖❈P✮ ✐♥❝❧✉❞❡s ◗❈◗P

❛✉①✐❧✐❛r② ✈❛r✐❛❜❧❡ ✐♥ ◗❈◗P tr❛♥s❢♦r♠s ❝♦♥str❛✐♥ts t♦ ❝♦♥✐❝

✹ s❡♠✐❞❡✜♥✐t❡ ♣r♦❣r❛♠♠✐♥❣ ✭❙❉P✮ ✐♥❝❧✉❞❡s ❙❖❈P

s❡♠✐❞❡✜♥✐t❡ ♠❛tr✐① ❝♦♥str❛✐♥ts ❣❡♥❡r❛❧✐s❡ s❡❝♦♥❞✲♦r❞❡r ❝♦♥✐❝

❊①❛♠♣❧❡ ✭▼❡❛♥✲✈❛r✐❛♥❝❡ ♦♣t✐♠✐s❛t✐♦♥✮

❇♦②❞ ❛♥❞ ❱❛♥❞❡♥❜❡r❣❤❡ ✭✷✵✵✹✱ ♣♣✳✶✺✺✲✶✺✻✮ ❞✐s❝✉ss ▼❛r❦♦✇✐t③ ✭✶✾✺✷✮ ♣♦rt❢♦❧✐♦ s❡❧❡❝t✐♦♥ ❛s ❛ q✉❛❞r❛t✐❝ ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠✳

✶✻✽ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦♥str❛✐♥❡❞ ♦♣t✐♠✐s❛t✐♦♥

❚❤❡ ▲♦r❡♥t③ ✭s❡❝♦♥❞✲♦r❞❡r✮ ❝♦♥❡

KM ≡

• ② ∈ RM

(y✶, . . . , yM−✶)′ ≤ yM

• K ✐s t❤❡ ▲♦r❡♥t③✱ ✐❝❡✲❝r❡❛♠✱

♥♦r♠ ♦r s❡❝♦♥❞✲♦r❞❡r ❝♦♥❡ ✇❤✐❧❡ t❤❡ ❝♦♥❡ ✐ts❡❧❢ ✐s ✉♥✐q✉❡✱ ❝♦♥str❛✐♥ts ❝❛♥ ❜❡ ✢❡①✐❜❧② ♣♦s❡❞ ❛s ❆i③ + ❜i ≤ ❢ ′

i ③ + di

❢♦r i = ✶, . . . , m

✶✻✾ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❈♦♥str❛✐♥❡❞ ♦♣t✐♠✐s❛t✐♦♥

❚❤❡ s❡♠✐❞❡✜♥✐t❡ ❝♦♥❡

SM

+ ≡ {❙ ✵}✱ ❙ ∈ RM×M ❛♥❞ ✵ ❞❡♥♦t❡s P❙❉

✐s ❛ ❝♦♥❡❄

❡✳❣✳ r❡♣r❡s❡♥t ❙ ≡ x y y z

• ❛s (x, y, z) ∈ R✸

P❙❉ ⇔ x ≥ ✵, z ≥ ✵, xz ≥ y✷ ❝❛♥ ✢❡①✐❜❧② ✇r✐t❡ ❝♦♥str❛✐♥ts ❋✵ +

n

• i=✶

❋izi ✵ s♦♠❡ ❙❉P s♦❧✈❡rs ❛r❡ ❧✐st❡❞ ❤❡r❡

✶✼✵ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❚❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❛♣♣r♦❛❝❤

❚❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❛♣♣r♦❛❝❤

▼❡✉❝❝✐ ✭✷✵✵✺✱ ➓✻✳✶✮ ❝♦♥t❛✐♥s ♣❡r❤❛♣s ♦♥❧② ♥♦♥✕tr✐✈✐❛❧ ♣♦rt❢♦❧✐♦ ♦♣t✐♠✐s❛t✐♦♥ ❡①❛♠♣❧❡ t❤❛t ❝❛♥ ❜❡ ❛♥❛❧②t✐❝❛❧❧② s♦❧✈❡❞ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❡✈❡♥ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥s ❝❛♥♥♦t ❜❡ ❣✉❛r❛♥t❡❡❞ ♦✉ts✐❞❡ ♦❢ ❝♦♥✈❡① ♣r♦❣r❛♠♠✐♥❣

S ♦r ❝♦♥str❛✐♥ts ❛r❡ ♥♦t ❝♦♥❝❛✈❡ q✉❛♥t✐❧❡ ♦r ❝❡rt❛✐♥t② ❡q✉✐✈❛❧❡♥t ❢❛✐❧❀ s♣❡❝tr❛❧ ♣❛ss

❡✈❡♥ ✇❤❡♥ t❤❡ ♣r♦❜❧❡♠ ✐s ♦♥❡ ♦❢ ❛ ❝♦♥✈❡① ♣r♦❣r❛♠♠✐♥❣✱ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st ♠❛② ❜❡ ♣r♦❤✐❜✐t✐✈❡ ❤❡r❡✱ ♣r❡s❡♥t ♠❡❛♥✕✈❛r✐❛♥❝❡ ❛♣♣r♦❛❝❤

✉s❡ ❞❛t❡s ❜❛❝❦ t♦ ▼❛r❦♦✇✐t③ ✭✶✾✺✷✮ ❡①tr❡♠❡❧② ♣♦♣✉❧❛r ❝♦♠♣✉t❛t✐♦♥❛❧❧② tr❛❝t❛❜❧❡

✶✼✷ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❚❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❛♣♣r♦❛❝❤

❚❤❡ ❣❡♦♠❡tr② ♦❢ ❛❧❧♦❝❛t✐♦♥ ♦♣t✐♠✐s❛t✐♦♥

t❤❡ ✐♥❞✐❝❡s ♦❢ s❛t✐s❢❛❝t✐♦♥ ❝♦♥s✐❞❡r❡❞ ❤❡r❡ ❛r❡ ❧❛✇ ✐♥✈❛r✐❛♥t t❤✉s✱ t❤❡② ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ Ψα ❞✐str✐❜✉t✐♦♥ ♦❢ Ψα✱ ✐♥ t✉r♥✱ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❡r♠s ♦❢ ♠♦♠❡♥ts

❝❡rt❛✐♥t② ❡q✉✐✈❛❧❡♥t✿ ❚❛②❧♦r ❡①♣❛♥❞✐♥❣ u (·) ②✐❡❧❞s ♠♦♠❡♥ts q✉❛♥t✐❧❡s ♦r s♣❡❝tr❛❧ ✐♥❞✐❝❡s✿ ❈♦r♥✐s❤✲❋✐s❤❡r ❡①♣❛♥s✐♦♥

t❤❡r❡❢♦r❡✱ ✐❢ S (α) ✐s ❛♥ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥✱ S (α) ≡ H (E {Ψα} , CM✷ {Ψα} , CM✸ {Ψα} , . . .) ✇❤❡r❡ CMk ✐s t❤❡ kt❤ ❝❡♥tr❛❧ ♠♦♠❡♥t ♦❢ Ψα ✐s♦✲s❛t✐s❢❛❝t✐♦♥ s✉r❢❛❝❡s t❤❡r❡❢♦r❡ ❧✐✈❡ ✐♥ (∞ − ✶)✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ♠♦♠❡♥ts ✇✐t❤✐♥ t❤✐s✱ t❤♦✉❣❤✱ α ∈ RN s♣❛♥s ❛ s✉❜s♣❛❝❡ ✇✐t❤✐♥ t❤❛t s✉❜s♣❛❝❡✱ t❤❡ ❝♦♥str❛✐♥t t❤❛t α ∈ C ✐s ❢✉rt❤❡r r❡str✐❝t✐✈❡ s♦❧✈✐♥❣ t❤❡ ♣r♦❜❧❡♠ ✐s ✜♥❞✐♥❣ t❤❡ ♣♦✐♥t ✐♥ t❤❛t ✜♥❛❧ s✉❜s♣❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❤✐❣❤❡st ❧❡✈❡❧ ♦❢ s❛t✐s❢❛❝t✐♦♥

✶✼✸ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❚❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❛♣♣r♦❛❝❤

❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥✿ t❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❢r❛♠❡✇♦r❦

✐♥st❡❛❞ ♦❢ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ✈❡rs✐♦♥ ♦❢ S (α)✱ ❝♦♥s✐❞❡r S (α) ≈ ˜ H (E {Ψα} , Var {Ψα}) t❤✐s ✇♦✉❧❞ ❝❧❡❛r❧② ❜❡ ❡❛s✐❡r t♦ s♦❧✈❡✱ ✇❤❡♥ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❣♦♦❞ ❛❧❧ S (α) ❝♦♥s✐❞❡r❡❞ ❛❜♦✈❡ ❛r❡ ❝♦♥s✐st❡♥t ✇✐t❤ st♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡

❣✐✈❡♥ s♦♠❡ ✜①❡❞ Var {Ψα}✱ ❤✐❣❤❡r E {Ψα} ♣r❡❢❡rr❡❞ ❢♦r ❛♥② ˜ H ❝❛♥♥♦t ❛ss✉♠❡ ❞✉❛❧✿ ❣✐✈❡♥ s♦♠❡ ✜①❡❞ E {Ψα}✱ ❧♦✇❡r Var {Ψα} ♣r❡❢❡rr❡❞ ❢♦r ❛♥② ˜ H❄

⇒ ♦♣t✐♠❛❧ ❛❧❧♦❝❛t✐♦♥ α∗ ❜❡❧♦♥❣s t♦ ✶✲♣❛r❛♠❡t❡r ❢❛♠✐❧② α (v)✱ ✇❤❡r❡ α (v) ≡ ❛r❣♠❛① α ∈ C Var {Ψα} = v E {Ψα} s♦❧✉t✐♦♥ ✐s ▼❛r❦♦✇✐t③✬ ♠❡❛♥✕✈❛r✐❛♥❝❡ ❡✣❝✐❡♥t ❢r♦♥t✐❡r

✶✼✹ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❚❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❛♣♣r♦❛❝❤

❆ t✇♦✲st❡♣ ❛♣♣r♦❛❝❤ t♦ t❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❢r♦♥t✐❡r

✶ ❝♦♠♣✉t❡ t❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❡✣❝✐❡♥t ❢r♦♥t✐❡r✱

α (v) ≡ ❛r❣♠❛① α ∈ C Var {Ψα} = v E {Ψα}

✷ ♣❡r❢♦r♠ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s❡❛r❝❤✱

α∗ ≡ α (v∗) ≡ ❛r❣♠❛①

v≥✵

S (α (v)) Ψα ≡ α′▼ ⇒ E {Ψ} = α′E {▼} , Var {Ψ} = α′Cov {▼} α✱ t❤✉s✱ ❝❛♥ ✇r✐t❡ ✜rst st❡♣ ✐♥ t❡r♠s ♦❢ t❤❡ ❤♦r✐③♦♥ ♠❛r❦❡t ✈❡❝t♦r✱ ▼✿ α (v) ≡ ❛r❣♠❛① α ∈ C α′Cov {▼} α = v ≥ ✵ α′E {▼} ✭✇❤② ❞♦ ✇❡ ❞♦ t❤✐s❄✮

✶✼✺ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❆♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ♣r♦❜❧❡♠

❆ s✐♥❣❧❡ ❛✣♥❡ ❝♦♥str❛✐♥t✿ ❛❧❧♦❝❛t✐♦♥ s♣❛❝❡

α✷ αN α✶ ❞ ′α = c αMV αSR

❊①❛♠♣❧❡

C : α′♣T = wT ❧❡t C ❜❡ ❛♥ ❛✣♥❡ ❝♦♥str❛✐♥t C : ❞ ′α = c s✳t✳ ❞, E {▼} ♥♦t ❝♦❧❧✐♥❡❛r ❢r♦♥t✐❡r t❤❡♥ ❛ s❡♠✐✲❧✐♥❡ ♦♥ (N − ✶)✲❞ ❝♦♥str❛✐♥t t✇♦✲❢✉♥❞ s❡♣❛r❛t✐♦♥ t❤❡♦r❡♠✿ ❢r♦♥t✐❡r ✐s ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ MV , SR ♣♦rt❢♦❧✐♦s

✶ ✇❤② ✐s ❢r♦♥t✐❡r ❛ ❧✐♥❡❄ ✷ ✇❤② ❜❡❣✐♥ ❛t αMV ❄ ✸ ✇❤② ❞♦tt❡❞ ❛❜♦✈❡ αSR❄ ✶✼✼ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ❆♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ♣r♦❜❧❡♠

❆ s✐♥❣❧❡ ❛✣♥❡ ❝♦♥str❛✐♥t✿ ♠❡❛♥✲✈❛r✐❛♥❝❡ s♣❛❝❡

αMV αSR E {Ψα} Var {Ψα}

❊①❛♠♣❧❡

C : α′♣T = wT ❛♣♣r♦①✐♠❛t✐♥❣ S (α) ❜② ✶st t✇♦ ♠♦♠❡♥ts

✶ ✇❤② ✐s ❢r♦♥t✐❡r ❛ ♣❛r❛❜♦❧❛❄ ✷ ✇❤② ❜❡❣✐♥ ❛t αMV ❄

✇❤❛t ✐s αMV ❄

✸ ✇❤② ❞♦tt❡❞ ❛❜♦✈❡ αSR❄

✇❤❛t ✐s αSR❄ ❋✐❣ ✻✳✶✶ ❞❡♣✐❝ts ✐♥ (E, Sd) s♣❛❝❡ SR (α) ≡ E {Ψα} Sd {Ψα}

❝❛♥ ❛♥❛❧②s❡ ♠❛r❦❡t ♥❡✉tr❛❧ s♣❡❝✐❛❧ ❝❛s❡ ❜② wT = ✵

✶✼✽ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s P✐t❢❛❧❧s ♦❢ t❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❢r❛♠❡✇♦r❦

▼❱ ❛s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥

r❡♠❡♠❜❡r✱ ❛r❡ ❛♣♣r♦①✐♠❛t✐♥❣ S (α) ❜② S (α) ≈ ˜ H (E {Ψα} , Var {Ψα}) ✇❤❡r❡ Ψα ≡ α′▼ ❛♣♣r♦①✐♠❛t✐♦♥ ❡①❛❝t ✐✛ S (α) ♦♥❧② ❞❡♣❡♥❞s ♦♥ ✶st t✇♦ ♠♦♠❡♥ts

♣r❡❢❡r❡♥❝❡s✿ ✐✛ S (α) ✐s ❝❡rt❛✐♥t② ❡q✉✐✈❛❧❡♥t ✇✐t❤ q✉❛❞r❛t✐❝ ✉t✐❧✐t② u (ψ) ≡ ψ − ✶ ✷γ ψ✷, t❤❡♥ ˜ H = H ❢♦r ❛♥② ▼ ♠❛r❦❡ts✿ ✐✛ ▼ ∼ El (µ, Σ, gN)✱ t❤❡♥ ˜ H = H ❢♦r ❛♥② S (·)

❤♦✇ ✇❡❧❧ ❞♦ t❤❡ ✜rst t✇♦ ♠♦♠❡♥ts ❝❛♣t✉r❡ ②♦✉r ♣❛rt✐❝✉❧❛r ♣r♦❜❧❡♠❄

❝❛♥ ✇❡ ❡①t❡♥❞ t❤✐s ♠❡t❤♦❞♦❧♦❣② t♦ ✜rst t❤r❡❡✱ ❢♦✉r ♠♦♠❡♥ts❄

✶✽✵ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s P✐t❢❛❧❧s ♦❢ t❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❢r❛♠❡✇♦r❦

❚❤❡ ❞✉❛❧ ❢♦r♠✉❧❛t✐♦♥✿ ❝❤❡❝❦ ❝❛s❡✲❜②✲❝❛s❡

E {Ψα} Var {Ψα}

❊①❛♠♣❧❡

◆❡t ♣r♦✜ts❄

r♦❜✉st ♦♣t✐♠✐s❛t✐♦♥

✇❤❡♥ ❝❛♥ ✇❡ ✇r✐t❡ α (v) ≡ ❛r❣♠❛① α ∈ C Var {Ψα} = v E {Ψα} . . .

✶ ❛s t❤❡ ✐♥❡q✉❛❧✐t② ❝♦♥str❛✐♥❡❞

α (v) ≡ ❛r❣♠❛① α ∈ C Var {Ψα} ≤ v E {Ψα}?

✷ ❛s t❤❡ ❞✉❛❧ ❢♦r♠✉❧❛t✐♦♥

α (e) ≡ ❛r❣♠✐♥ α ∈ C E {Ψα} ≥ e Var {Ψα}?

✶✽✶ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s P✐t❢❛❧❧s ♦❢ t❤❡ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❢r❛♠❡✇♦r❦

▲❡❝t✉r❡ ✾ ❡①❡r❝✐s❡s

▼❡✉❝❝✐ ❡①❡r❝✐s❡s

♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✽✳✶✳✸ ✭♥♦♥✲▼❆❚▲❆❇ ❝♦♠♣♦♥❡♥t✮ ▼❆❚▲❆❇✿ ✽✳✶✳✶✱ ✽✳✶✳✷✱ ✽✳✶✳✸ ✭▼❆❚▲❆❇ ❝♦♠♣♦♥❡♥t✮✱ ✽✳✶✳✹ ✭❤✐♥t✿ ②♦✉ ♥❡❡❞ ❛ ✇♦r❦❛r♦✉♥❞ ❢♦r ❜❧s♣r✐❝❡✮

♣r♦❥❡❝t✿ ❣✐✈❡♥ ❛♥ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ ♦❢ ✻ ❋❡❜ ✷✵✶✼

✶

❢♦r ②♦✉r ♣♦rt❢♦❧✐♦ ✭♦r r❡❞✉❝❡❞ ❞✐♠❡♥s✐♦♥ ✈❡rs✐♦♥✮✱ ✐♠♣❧❡♠❡♥t t❤❡ t✇♦✲st❡♣ ♠❡❛♥✲✈❛r✐❛♥❝❡ s❡❛r❝❤ ♣r♦❝❡❞✉r❡✱ ✜rst ✐❞❡♥t✐❢②✐♥❣ t❤❡ ❧♦❝✉s ♦❢ α (v)✱ ❛♥❞ t❤❡♥ α∗

✷

t❤✐♥❦ ❛❜♦✉t ❛♥❞ ❡①♣❧❛✐♥ ✇❤② t❤❡ r❡s✉❧t✐♥❣ α∗ ✐s t❤❡ ♦♣t✐♠❛❧ ♣♦rt❢♦❧✐♦ ✭❣✐✈❡♥ ❛ ♣❛rt✐❝✉❧❛r ❢♦r❡❝❛st ♠❡t❤♦❞✮

✶✽✷ ✴ ✷✷✹

❊st✐♠❛t✐♥❣ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❇❛②❡s✐❛♥ ❡st✐♠❛t✐♦♥

❇❛②❡s✐❛♥ ❡st✐♠❛t✐♦♥

❝❧❛ss✐❝❛❧ ❡st✐♠❛t✐♦♥✿ ❢r♦♠ ❞❛t❛ t♦ ♣❛r❛♠❡t❡rs iT ≡ {①✶, . . . , ①T} → ˆ θ

❜✉t ❞✐✛❡r❡♥t r❡❛❧✐s❛t✐♦♥s ❢r♦♠ t❤❡ s❛♠❡ ❉●P → ❞✐✛❡r❡♥t ˆ θ

❇❛②❡s✐❛♥ ❡st✐♠❛t✐♦♥

❡①♣❡r✐❡♥❝❡✱ eC ✭❛♥❞ ❝♦♥✜❞❡♥❝❡ ✐♥ ✐t✮ ✐♠♣❧② ♣r✐♦r ❜❡❧✐❡❢s✱ fpr (θ✵) ❝♦♠❜✐♥✐♥❣ t❤✐s ✇✐t❤ ˆ θ ❢r♦♠ iT ②✐❡❧❞s ❛ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ iT, eC → fpo (θ) ❝❛♥ ✈✐❡✇ ♣r✐♦r ❜❡❧✐❡❢s ❛s r❡✢❡❝t✐♥❣ C ♣s❡✉❞♦✲♦❜s❡r✈❛t✐♦♥s C → ∞ ⇒ fpo (θ) → θ✵; T → ∞ ⇒ fpo (θ) → ˆ θ;

❝❧❛ss✐❝❛❧✲❡q✉✐✈❛❧❡♥t ❡st✐♠❛t♦rs ✉s❡ ❧♦❝❛t✐♦♥ ♣❛r❛♠❡t❡r ✭❡✳❣✳ ❡①♣❡❝t❡❞ ✈❛❧✉❡✱ ♠♦❞❡✮ t♦ s✉♠♠❛r✐s❡ fpo

❇❛②❡s✲❙t❡✐♥ s❤r✐♥❦❛❣❡ ❡st✐♠❛t♦rs ✇❤❡♥ s❤r✐♥❦ ♦♥t♦ ♣r✐♦r ❜❡❧✐❡❢s

s❤r✐♥❦❛❣❡ ❡st✐♠❛t♦rs

♦❢t❡♥ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ❡①♣❡♥s✐✈❡ ❞✉❡ t♦ ✐♥t❡❣r❛t✐♦♥ ♠♦❞❡❧❧✐♥❣ ♣r✐♦rs ❛s ♥♦r♠❛❧✲✐♥✈❡rs❡✲❲✐s❤❛rt ♠❛② ❛✐❞ tr❛❝t❛❜✐❧✐t②

❲✐s❤❛rt ✶✽✹ ✴ ✷✷✹

❊st✐♠❛t✐♥❣ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❉❡t❡r♠✐♥✐♥❣ t❤❡ ♣r✐♦r

❉❡t❡r♠✐♥✐♥❣ t❤❡ ♣r✐♦r

t❤❡♦r② ♥✐❝❡✱ ❜✉t ✇❤♦ ❝❛♥ ❝♦♥✈❡rt t❤❡✐r ❜❡❧✐❡❢s ✐♥t♦ ❛ ❞✐str✐❜✉t✐♦♥❄

✶ ♣❡❛❦ t❤❡♥ t✇❡❛❦

♦❢t❡♥ s♣❡❝✐❢② ❧♦❝❛t✐♦♥ ♣❛r❛♠❡t❡r ❢r♦♠ ❵♣❡❛❦✬ ♦❢ ♣r✐♦r ❜❡❧✐❡❢s t❤❡♥ ❵t✇❡❛❦✬ ❞✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r t♦ ✈❛r② ❝♦♥✜❞❡♥❝❡ ❧❡✈❡❧s

✷ ❛❧❧♦❝❛t✐♦♥ ✐♠♣❧✐❡❞ ♣❛r❛♠❡t❡rs

✐♥✈❡st♦rs ♦❢t❡♥ ❤❛✈❡ ❛ ❜❡tt❡r ✐❞❡❛ ♦❢ t❤❡✐r ♣r❡❢❡rr❡❞ ♣♦rt❢♦❧✐♦✱ α✱ t❤❛♥ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ♠❛r❦❡t ♣❛r❛♠❡t❡rs✱ θ ✈✐❡✇ ♣r❡❢❡rr❡❞ ♣♦rt❢♦❧✐♦ ❛s s♦❧✈✐♥❣ α (θ) ≡ ❛r❣♠❛①

α∈C

Sθ (α) ✐❢ θ✬s ❞✐♠❡♥s✐♦♥ ♠♦r❡ t❤❛♥ N✱ ♥❡❡❞ ❢✉rt❤❡r r❡str✐❝t✐♦♥s t♦ ✐♥✈❡rt ❢♦r θ

✸ ♣r✐♦r ❝♦♥str❛✐♥❡❞ ❧✐❦❡❧✐❤♦♦❞ ♠❛①✐♠✐s❛t✐♦♥

❡①✐st✐♥❣ α ✐♠♣❧✐❡s ˜ Θ ⊂ Θ✱ s✉❜s❡t ♦❢ ♣❛r❛♠❡t❡rs ❝♦♥s✐st❡♥t ✇✐t❤ α ∈ C ✉s❡ ▼▲ ♦♥ iT t♦ ❡st✐♠❛t❡ ♣r✐♦rs ✇✐t❤✐♥ ˜ Θ

✶✽✻ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❆❧❧♦❝❛t✐♦♥s ❛s ❞❡❝✐s✐♦♥s

❖♣♣♦rt✉♥✐t② ❝♦st ♦❢ s✉❜♦♣t✐♠❛❧ ❛❧❧♦❝❛t✐♦♥s

❛♥ ♦♣t✐♠❛❧ ❛❧❧♦❝❛t✐♦♥ s♦❧✈❡s α∗ ≡ ❛r❣♠❛①

α∈C

S (α) t❤❡ ♦♣♣♦rt✉♥✐t② ❝♦st ♦❢ ❛ ❣❡♥❡r✐❝ ❛❧❧♦❝❛t✐♦♥ α ✐s OC (α) ≡ S (α∗) − S (α) ≥ ✵ ✭❢♦r ❡①♣♦s✐t✐♦♥❛❧ ❝❧❛r✐t②✱ ✐❣♥♦r✐♥❣ ❝♦sts ♦❢ ❝♦♥str❛✐♥t ✈✐♦❧❛t✐♦♥✮ ❛s s❛t✐s❢❛❝t✐♦♥ ❢r♦♠ ❛♥② α ❞❡♣❡♥❞s ♦♥ t❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡rs θ → ❳ θ

T+τ → Pθ T+τ ⇒

• α, Pθ

T+τ

• → Ψθ

α → Sθ (α)

s♦ ❞♦❡s t❤❡ ♦♣♣♦rt✉♥✐t② ❝♦st OC θ (α) ≡ Sθ (α∗ (θ)) − Sθ (α) ≥ ✵ ❛s ❣❡♥❡r✐❝ ❛❧❧♦❝❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ ❞❛t❛✱ iT✱ s♦ ❞♦❡s ♦♣♣♦rt✉♥✐t② ❝♦st OC θ (α [iT]) ≡ Sθ (α∗ (θ)) − Sθ (α [iT]) ≥ ✵ ❜✉t ✉♥♦❜t❛✐♥❛❜❧❡ Sθ (α∗ (θ)) ❞♦❡s ♥♦t ✭✇❤②❄✮

✶✽✽ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❆❧❧♦❝❛t✐♦♥s ❛s ❞❡❝✐s✐♦♥s

❖♣♣♦rt✉♥✐t② ❝♦st ❛s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡

iT ✐s ❛ r❡❛❧✐s❛t✐♦♥ ♦❢ r❛♥❞♦♠ I θ

T ≡

• ❳ θ

✶ , . . . , ❳ θ T

• t❤✉s✱ ✐❢ ❛♥ ❛❧❧♦❝❛t✐♦♥ ✐s r❡s✉❧t ♦❢ ❛ ❞❡❝✐s✐♦♥ r✉❧❡✱ α
• I θ

T

• ✐s ❛♥ r✈

✐♥ t✉r♥✱ t❤❡ ♦♣♣♦rt✉♥✐t② ❝♦st ✐ts❡❧❢ ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ OC θ α

• I θ

T

• ≡ Sθ (α∗ (θ)) − Sθ

α

• I θ

T

• ≥ ✵

❝❛♥ ♥♦✇ str❡ss t❡st ❛♥ ❛❧❧♦❝❛t✐♦♥ ✭♠❛♣♣✐♥❣ ❢r♦♠ t❤❡ iT✮

❤♦✇ ❞♦❡s ❖❈ ✈❛r② ♦✈❡r Θ✱ s✉❜s❡t ❡①♣❡❝t❡❞ t♦ ❝♦♥t❛✐♥ t❤❡ tr✉❡ θ❄ ✐❞❡❛❧❧②✱ ✇❛♥t ❖❈ ❧♦✇ ❢♦r ❛❧❧ θ ∈ Θ

▼❡✉❝❝✐ ❞♦❡s ♥♦t ❛❣❣r❡❣❛t❡ t❤✐s ✐♥t♦ ❛ s✐♥❣❧❡ ♥✉♠❜❡r

S (α) ❛❣❣r❡❣❛t❡s Ψα✱ ❜✉t ✳ ✳ ✳ ✏♠♦❞❡❧✐♥❣ t❤❡ ✐♥✈❡st♦r✬s ❛tt✐t✉❞❡ t♦✇❛r❞ ❡st✐♠❛t✐♦♥ r✐s❦ ✐s ❛♥ ❡✈❡♥ ❤❛r❞❡r t❛s❦ t❤❛♥ ♠♦❞❡❧✐♥❣ ❤✐s ❛tt✐t✉❞❡ t♦✇❛r❞ r✐s❦✑ ✐s t❤✐s ✇❤❛t t❤❡ ❛♠❜✐❣✉✐t② ❧✐t❡r❛t✉r❡ ✭❊♣st❡✐♥ ❛♥❞ ❙❝❤♥❡✐❞❡r✱ ✷✵✶✵✮ s❡❡❦s t♦ ❞♦❄

✶✽✾ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ Pr✐♦r ❛❧❧♦❝❛t✐♦♥

❆❧❧♦❝❛t✐♥❣ ♦♥ t❤❡ ❜❛s✐s ♦❢ ♣r✐♦rs ♦♥❧②

❝♦♥s✐❞❡r t❤❡ ♣r✐♦r ❛❧❧♦❝❛t✐♦♥ r✉❧❡ αp [iT] ≡ α ✇❤❡r❡ α ✐s ❛ ✜①❡❞ ♣♦rt❢♦❧✐♦ t❤❡♥ t❤❡ ♦♣♣♦rt✉♥✐t② ❝♦st ❜❡❝♦♠❡s ❞❡t❡r♠✐♥✐st✐❝✱ ❣✐✈❡♥ θ OC θ αp

• I θ

T

• ≡ Sθ (α∗ (θ)) − Sθ (αp) ≥ ✵

❢✉rt❤❡r✱ OC θ (αp) ✐s ❣❡♥❡r❛❧❧② ❧❛r❣❡

q✳✈✳ ❜✐❛s ♦❢ ❝♦♥st❛♥t ❡st✐♠❛t♦r

❊①❛♠♣❧❡ ✭❊q✉❛❧❧②✲✇❡✐❣❤t❡❞ ♣♦rt❢♦❧✐♦✮

❚❤❡ ❡q✉❛❧❧②✲✇❡✐❣❤t❡❞ ♣♦rt❢♦❧✐♦ ✐s ❞❡t❡r♠✐♥❡❞ ❡①❝❧✉s✐✈❡❧② ❜② ♣r✐♦r ✈✐❡✇s✳

✶✾✶ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❙❛♠♣❧❡ ❛❧❧♦❝❛t✐♦♥

❙❛♠♣❧❡✲❜❛s❡❞ ❛❧❧♦❝❛t✐♦♥

✉♥t✐❧ t❤✐s ✇❡❡❦✱ ❤❛✈❡ ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ❢r♦♠ ❞❛t❛✱ ˆ θ [iT] ❧❡t t❤❡ s❛♠♣❧❡ ❛❧❧♦❝❛t✐♦♥ ❜❡ αs [iT] ≡ α

• ˆ

θ [iT]

• ≡ ❛r❣♠❛①

α∈C

ˆ θ[iT ]

S

ˆ θ[iT ] (α)

t♦ ❡✈❛❧✉❛t❡ ♣❡r❢♦r♠❛♥❝❡✱ ❞♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ∀θ ∈ Θ ✭t❤❡ str❡ss t❡st s❡t✮

✶

❝♦♠♣✉t❡ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ Sθ (α∗ (θ))

✷

❣❡♥❡r❛t❡ ❛ ❞✐str✐❜✉t✐♦♥ ♦❢ iT✬s ⇒ ❞✐str✐❜✉t✐♦♥ ❢♦r ˆ θ

• I θ

T

• ✸

♣r♦❞✉❝❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❵♦♣t✐♠❛❧✬ ❛❧❧♦❝❛t✐♦♥s✱ ✐♥❞❡①❡❞ ❜② ˆ θ αs

• I θ

T

• ≡ α
• ˆ

θ

• I θ

T

• ≡ ❛r❣♠❛①

α∈C

ˆ θ[Iθ T]

S

ˆ θ[I θ

T ] (α)

✹

❝♦♠♣✉t❡ ❞✐str✐❜✉t✐♦♥ ♦❢ Sθ αs

• I θ

T

• ✱ ❣✐✈❡♥ ✐♥❞❡①❡❞ θ

✺

❝♦♠♣✉t❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ OC θ αs

• I θ

T

• ≡ Sθ (α∗ (θ)) − Sθ

αs

• I θ

T

• t❤✉s✱ ❢♦r ❡❛❝❤ θ ∈ Θ ❤❛✈❡ ❛ ❞✐str✐❜✉t✐♦♥ ♦❢ OC θ

αs

• I θ

T

• ✶✾✸ ✴ ✷✷✹

❊✈❛❧✉❛t✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❙❛♠♣❧❡ ❛❧❧♦❝❛t✐♦♥

❊✈❛❧✉❛t✐♥❣ t❤❡ s❛♠♣❧❡✲❜❛s❡❞ ❛❧❧♦❝❛t✐♦♥ ❛♣♣r♦❛❝❤

✐❢ ˆ θ ✐s ❛♥ ✉♥❜✐❛s❡❞ ❡st✐♠❛t♦r ♦❢ θ✱ t❤❡♥ ❜✉❧❦ ♦❢ OC θ αs

• I θ

T

• ❞✐str✐❜✉t✐♦♥ ✐s ❝❧♦s❡ t♦ ③❡r♦

❤♦✇❡✈❡r✱ αs ✐s ✐♥❡✣❝✐❡♥t ❞✉❡ t♦ s❡♥s✐t✐✈✐t② ♦❢ ♦♣t✐♠❛❧ ❛❧❧♦❝❛t✐♦♥ t♦ ✐♥❡✣❝✐❡♥❝② ✐♥ ˆ θ

✶

❛s s❛♠♣❧❡✲❜❛s❡❞ ❡st✐♠❛t♦rs ❛r❡ ✐♥❡✣❝✐❡♥t✱ ˆ θ

• I θ

T

• ✐s

✷

✐♥❡✣❝✐❡♥t ❡st✐♠❛t❡s ♦❢ ˆ θ

• I θ

T

• ♣r♦♣❛❣❛t❡ ❡st✐♠❛t✐♦♥ ❡rr♦r ✐♥t♦ ❡st✐♠❛t❡s

♦❢ s❛t✐s❢❛❝t✐♦♥✱ S ˆ

θ✱ t❤❡ ❝♦♥str❛✐♥ts✱ C ˆ θ✱ ❛♥❞ ✳ ✳ ✳

✸

t❤❡ ❝♦♠♣✉t❡❞ ♦♣t✐♠❛❧ ❛❧❧♦❝❛t✐♦♥ ✐ts❡❧❢✱ αs

t❤✉s✱ ❛❧❧♦❝❛t✐♦♥s ❝❛♥ ✈❛r② ❣r❡❛t❧② ✇✐t❤ t❤❡ ♣❛rt✐❝✉❧❛r ❤✐st♦r② ✉s❡❞ ❝❛♥ tr❛❞❡ ♦✛ ❜✐❛s ❛❣❛✐♥st ❡✣❝✐❡♥❝② ❜② ✉s✐♥❣ s❤r✐♥❦❛❣❡ ❡st✐♠❛t♦rs

s❤r✐♥❦❛❣❡ ❡st✐♠❛t♦rs ✶✾✹ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦

❖✈❡r✈✐❡✇

❛♣♣r♦❛❝❤ ❞❡s❝r✐❜❡❞ ♣r❡✈✐♦✉s❧②

✶

❡st✐♠❛t❡s ♠❛r❦❡t ❞✐str✐❜✉t✐♦♥

✷

✐♥♣✉ts t❤❡s❡ ❡st✐♠❛t❡s ✐♥t♦ ❛ ❝❧❛ss✐❝❛❧ ♦♣t✐♠✐s❡r

s♦ t❤❛t t❤❡ ♣❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥ ✐♥❡✣❝✐❡♥❝② ♣r♦♣❛❣❛t❡s t❤r♦✉❣❤ ♥♦✇ ❝♦♥s✐❞❡r ❛❧t❡r♥❛t✐✈❡s t❤❛t ❧✐♠✐t s❡♥s✐t✐✈✐t②

✶

❇❛②❡s✐❛♥ ❛❧❧♦❝❛t✐♦♥ t❤❛t s❤r✐♥❦s ♣❛r❛♠❡t❡rs ❡st✐♠❛t❡s t♦ ♣r✐♦rs ♦♥ θ

✷

❇❧❛❝❦✕▲✐tt❡r♠❛♥ ❛❧❧♦❝❛t✐♦♥ t❤❛t s❤r✐♥❦s ✇✐t❤ r❡s♣❡❝t t♦ ♠❛r❦❡t ✈✐❡✇s

t❤✐s ✷✵✶✹ ❛rt✐❝❧❡ ❞✐s❝✉ss❡s r♦❜♦❛❞✈✐s♦rs✬ ✉s❡ ♦❢ ❇❧❛❝❦✲▲✐tt❡r♠❛♥

✸

▼✐❝❤❛✉❞ r❡s❛♠♣❧✐♥❣ ✳ ✳ ✳ ❄

❛s ✇❡❧❧ ❛s ❛♣♣r♦❛❝❤❡s t❤❛t ❞♦♥✬t ❥✉st tr② t♦ ❧✐♠✐t s❡♥s✐t✐✈✐t②

✹

r♦❜✉st ❛❧❧♦❝❛t✐♦♥ ❞♦❡s♥✬t ❧✐♠✐t s❡♥s✐t✐✈✐t②✱ ❜✉t ♣✐❝❦s t♦ ❡♥s✉r❡ ❛❣❛✐♥st ❜❛❞ ♦✉t❝♦♠❡

✺

r♦❜✉st ❇❛②❡s✐❛♥ ❜❧❡♥❞s r♦❜✉st ✇✐t❤ ❇❛②❡s✐❛♥

✶✾✺ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❇❛②❡s✐❛♥ ❛❧❧♦❝❛t✐♦♥

❇❛②❡s✐❛♥ ❛❧❧♦❝❛t✐♦♥

❛s ❞♦♥✬t ❦♥♦✇ tr✉❡ θ✱ ❝❛♥ ♥❡✈❡r ✐♠♣❧❡♠❡♥t ♦♣t✐♠❛❧ ❛❧❧♦❝❛t✐♦♥ α (θ) ≡ ❛r❣♠❛①

α∈Cθ Sθ (α)

❜✉t ❝❛♥ ✐♠♣❧❡♠❡♥t ❝❧❛ss✐❝❛❧✕❡q✉✐✈❛❧❡♥t ❇❛②❡s✐❛♥ ❛❧❧♦❝❛t✐♦♥ ❞❡❝✐s✐♦♥ αce [iT, eC] ≡ α

• ˆ

θce [iT, eC]

• ≡

❛r❣♠❛①

α∈C

ˆ θce[iT ,eC ]

S

ˆ θce[iT ,eC ] (α)

✇❤❡r❡ ˆ θce [iT, eC] ✐s ♣♦st❡r✐♦r ❧♦❝❛t✐♦♥ ♣❛r❛♠❡t❡r ❛s ✇✐t❤ s❛♠♣❧❡✕❜❛s❡❞✱ ❡✈❛❧✉❛t❡ ❈❊❇❆ ♦✈❡r ❛❧❧ θ ∈ Θ ✭❞♦♠❛✐♥ ♦❢ fpo✮

✶

❝♦♠♣✉t❡ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ Sθ (α∗ (θ))

✷

❣❡♥❡r❛t❡ ❛ ❞✐str✐❜✉t✐♦♥ ♦❢ iT✬s ⇒ ❞✐str✐❜✉t✐♦♥ ❢♦r ˆ θce

• I θ

T, eC

• ✸

♣r♦❞✉❝❡ αce

• I θ

T, eC

• ✱ ❞✐str✐❜✉t✐♦♥ ♦❢ ❵♦♣t✐♠❛❧✬ ❛❧❧♦❝❛t✐♦♥s ❣✐✈❡♥ ˆ

θce

✹

❝♦♠♣✉t❡ ❞✐str✐❜✉t✐♦♥ ♦❢ Sθ αce

• I θ

T, eC

• ✱ ❣✐✈❡♥ ✐♥❞❡①❡❞ θ

✺

❝♦♠♣✉t❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ OC θ αce

• I θ

T, eC

• r❡❧❛t✐✈❡ t♦ αs✱ ♠✐♥✐♠✐s❡s✱ t✐❣❤t❡♥s OC ❡s♣❡❝✐❛❧❧② ✇❤❡r❡ ♣r✐♦r str♦♥❣❡st

✶✾✼ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❇❧❛❝❦✲▲✐tt❡r♠❛♥ ❛❧❧♦❝❛t✐♦♥

❇❧❛❝❦✲▲✐tt❡r♠❛♥ ❛❧❧♦❝❛t✐♦♥

❛s ❞♦♥✬t ❦♥♦✇ tr✉❡ θ✱ ❝❛♥ ♥❡✈❡r ✐♠♣❧❡♠❡♥t ♦♣t✐♠❛❧ ❛❧❧♦❝❛t✐♦♥ α (θ) ≡ ❛r❣♠❛①

α∈Cθ Sθ (α)

❛s ❈❊❇❆✱ ❇▲ ✉s❡s ❇❛②❡s t♦ ❧✐♠✐t s❡♥s✐t✐✈✐t② t♦ θ✬s ✐♥❡✣❝✐❡♥t ❡st✐♠❛t✐♦♥ ❈❊❇❆ s❤r✐♥❦s ❡st✐♠❛t❡s ♦❢ ♠❛r❦❡t ♣❛r❛♠❡t❡rs✱ θ✱ t♦ t❤❡✐r ♣r✐♦rs ❇▲ s❤r✐♥❦s ❡st✐♠❛t❡s ♦❢ ♠❛r❦❡t ❞✐str✐❜✉t✐♦♥✱ s❛② ❳✱ t♦ t❤❡✐r ♣r✐♦rs

✶

❣✐✈❡♥ s♦♠❡ ♠❛r❦❡t r✈ ❳✱ q✉❛♥ts ❞❡t❡r♠✐♥❡ ❛ ❞✐str✐❜✉t✐♦♥ f❳

✷

❡①♣❡r✐❡♥❝❡❞ ✐♥✈❡st♦r ♣r♦✈✐❞❡s ✈✐❡✇✱ ✈

✈ s❡❡♥ ❛s r❡❛❧✐s❛t✐♦♥ ♦❢ r✈ ❱ ✭❡❧s❡ ❝♦♠♣❧❡t❡ s❤r✐♥❦❛❣❡✮ ❱ |g (①) ✐s ✐♥✈❡st♦r✬s ✈✐❡✇ ❣✐✈❡♥ ♠♦❞❡❧ ♣r❡❞✐❝t✐♦♥✱ ❡✳❣✳ ❱ |① ∼ N (①, Ω′) g (①) ❛❧❧♦✇s t❤❡ ✐♥✈❡st♦r✬s ✈✐❡✇s t♦ ❞❡♣❡♥❞ ♦♥ ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♠❛r❦❡t

✸

❇❛②❡s✬ r✉❧❡ ❝♦♠♣✉t❡s ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥ f❳|✈ (①) = f❱ |g(①) (✈) f❳ (①)

• f❱ |g(①′) (✈) f❳ (①′) d①′

✹

❇❧❛❝❦✲▲✐tt❡r♠❛♥ ❛❧❧♦❝❛t✐♦♥ ❞❡❝✐s✐♦♥ ❞❡♣❡♥❞s ♦♥ iT ✐✛ q✉❛♥t ♠♦❞❡❧ ❞♦❡s αBL [✈] ≡ ❛r❣♠❛①

α∈C✈ S✈ (α)

✶✾✾ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❇❧❛❝❦✲▲✐tt❡r♠❛♥ ❛❧❧♦❝❛t✐♦♥

❊①❛♠♣❧❡ ✭▲✐♥❡❛r ❡①♣❡rt✐s❡ ♦♥ ♥♦r♠❛❧ ♠❛r❦❡ts ✭▼❡✉❝❝✐✱ ✷✵✶✵✮✮

✶ st♦❝❦ ✐♥❞✐❝❡s ❢♦r ■t❛❧②✱ ❙♣❛✐♥✱ ❙✇✐t③❡r❧❛♥❞✱ ❈❛♥❛❞❛✱ ❯❙❆✱ ●❡r♠❛♥② ❛r❡

♠♦❞❡❧❧❡❞ ❛s ♥♦r♠❛❧✱ ❳ ∼ N (µ, Σ)

✷ ✐♥✈❡st♦r ♣r♦✈✐❞❡s ♣♦✐♥t ❡st✐♠❛t❡s ♦♥ ❛r❡❛s ♦❢ ❡①♣❡rt✐s❡✱ ✈

t✇♦ ✈✐❡✇s✿ ❛ ✶✷✪ ❙♣❛♥✐s❤ ❣❛✐♥✱ ❛♥❞ ✶✵✪ ●❡r♠❛♥ ♦✉t♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ❯❙ ✐♥✈❡st♦r✬s ❡①♣❡rt✐s❡ ✐s ❧✐♥❡❛r✱ g (①) = P①

P ✐s K × N ♣✐❝❦ ♠❛tr✐①✱ r❡♣r❡s❡♥t✐♥❣ K ✈✐❡✇s kt❤ r♦✇ ✐s N✲✈❡❝t♦r ❝♦rr❡s♣♦♥❞✐♥❣ t♦ kt❤ ✈✐❡✇ ❛ ✈✐❡✇ ♦♥ ❙♣❛✐♥✱ ❛♥❞ ♦♥❡ ♦♥ ❯❙✲●❡r♠❛♥ r❡❧❛t✐✈❡ ♣❡r❢♦r♠❛♥❝❡ ❛r❡ P = ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✶ −✶

• ❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ✈✐❡✇s ✐s ♥♦r♠❛❧✱ ❱ |P① ∼ N (P①, Ω)

✸ t❤❡ ♣♦st❡r✐♦r ♠❛r❦❡t ✈❡❝t♦r ❣✐✈❡♥ t❤❡ ✈✐❡✇ ✐s ❳|✈ ∼ N (µBL, ΣBL)✳ ✷✵✵ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❘♦❜✉st ❛❧❧♦❝❛t✐♦♥

❘♦❜✉st ❛❧❧♦❝❛t✐♦♥

♥♦✇✱ ❞♦♥✬t tr② t♦ r❡❞✉❝❡ s❡♥s✐t✐✈✐t② t♦ ✐♥❡✣❝✐❡♥t ❡st✐♠❛t✐♦♥ ♦❢ θ ✐♥st❡❛❞✱ ♠♦st ❝♦♥s❡r✈❛t✐✈❡ ❛♣♣r♦❛❝❤✿ ♠✐♥✐♠✐s❡ t❤❡ ♠❛①✐♠✉♠ OC ♦✈❡r t❤❡ str❡ss✕t❡st s❡t

✶ ✉s❡ iT t♦ ❞❡✜♥❡ r♦❜✉st♥❡ss s❡t✱ ˆ

Θ [iT]✱ s♠❛❧❧❡st Θ t❤❛t ❝♦♥t❛✐♥s tr✉❡ θ

✷ ❞❡✜♥❡ ❝♦♥str❛✐♥t s❡t t♦ ❡♥s✉r❡ ❛❧❧♦❝❛t✐♦♥ ❢❡❛s✐❜❧❡ ❢♦r ❛♥② θ ∈ Θ

C

ˆ Θ[iT ] ≡

• α ∈ Cθ∀θ ∈ ˆ

Θ [iT]

• ✸ t❤❡ r♦❜✉st ❛❧❧♦❝❛t✐♦♥ ❞❡❝✐s✐♦♥ t❤❡♥ ♠❛♣s ❢r♦♠ iT t♦ s♦❧✈❡

αr [iT] ≡ ❛r❣♠✐♥

α∈C

ˆ Θ[iT ]

♠❛① θ∈ ˆ

Θ[iT ]

• Sθ (α∗ (θ)) − Sθ (α)
• ❡✳❣✳ ③❡r♦✲s✉♠ ❣❛♠❡ ❛❣❛✐♥st ❡✈✐❧ ❞❡♠♦♥ ♣✐❝❦✐♥❣ t❤❡ ✇♦rst θ ∈ Θ

✷✵✸ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❘♦❜✉st ❛❧❧♦❝❛t✐♦♥

▼❡❛♥✲✈❛r✐❛♥❝❡ ❢r❛♠❡✇♦r❦ ❢♦r r♦❜✉st ❛❧❧♦❝❛t✐♦♥

♣r♦❤✐❜✐t✐✈❡❧② ❡①♣❡♥s✐✈❡ t♦ ✐♠♣❧❡♠❡♥t ♠✐♥♠❛① ❝♦♠♣✉t❛t✐♦♥❛❧❧②

t❤❡r❡❢♦r❡✱ ✉s❡ t✇♦✲st❡♣ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❢r❛♠❡✇♦r❦ ❛❣❛✐♥

❢✉rt❤❡r s✐♠♣❧✐❢②✐♥❣ ❛ss✉♠♣t✐♦♥s

✶

❝♦♥str❛✐♥ts ❞♦♥✬t ❞❡♣❡♥❞ ♦♥ Θ

✷

❝❛♥ ✇r✐t❡ ✈❛r✐❛♥❝❡ ❝♦♥str❛✐♥t ❛s Var {Ψα} ≤ v ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ➓✻✳✺✳✸✮

❞✉❛❧

❣✐✈❡♥ Θ✱ ✜rst st❡♣ t❤❡♥ ❜❡❝♦♠❡s αr (v) ≡ ❛r❣♠❛①

α

♠✐♥

µ∈ ˆ Θµ

α′µ s✳t✳

• α

∈ C ♠❛①Σ∈ ˆ

ΘΣ α′Σα

≤ v ❝❛r❡❢✉❧ ❝❤♦✐❝❡ ♦❢ ˆ Θ ❛❧❧♦✇s ♣r♦❜❧❡♠ t♦ ❜❡ ❝❛st ❛s ❙❖❈P

❊①❛♠♣❧❡ ✭❊❧❧✐♣t✐❝❛❧ ❡①♣❡❝t❛t✐♦♥s✱ ❦♥♦✇♥ ❝♦✈❛r✐❛♥❝❡s✮

✶ ❡❧❧✐♣t✐❝❛❧ ❡①♣❡❝t❛t✐♦♥s✿ ˆ

Θµ ≡

• µ s✳t✳ Ma✷ (µ, ♠, ❚) ≤ q✷

✷ ❦♥♦✇♥ ❝♦✈❛r✐❛♥❝❡s✿ ˆ

ΘΣ ≡ ˆ Σ

✷✵✹ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❘♦❜✉st ❇❛②❡s✐❛♥ ❛❧❧♦❝❛t✐♦♥

▲❡❝t✉r❡ ✶✵ ❡①❡r❝✐s❡s

▼❡✉❝❝✐ ❡①❡r❝✐s❡s

♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ▼❆❚▲❆❇✿ ✶✵✳✶✳✶

♣r♦❥❡❝t✿ ❝❛♥ ②♦✉ ❡①t❡♥❞ ❧❡❝t✉r❡ ✾✬s ❡①❡r❝✐s❡s t♦ ❤❛♥❞❧❡ ❡st✐♠❛t✐♦♥ r✐s❦❄

✷✵✻ ✴ ✷✷✹

❖♣t✐♠✐s✐♥❣ ❛❧❧♦❝❛t✐♦♥s ✇✐t❤ ❡st✐♠❛t✐♦♥ r✐s❦ ❘♦❜✉st ❇❛②❡s✐❛♥ ❛❧❧♦❝❛t✐♦♥

❚❤❡ ▼❡✉❝❝✐ ♠❛♥tr❛

✶ ❢♦r ❡❛❝❤ s❡❝✉r✐t②✱ ✐❞❡♥t✐❢② t❤❡ ✐✐❞ st♦❝❤❛st✐❝ t❡r♠s ✭➓✸✳✶✮ ✷ ❡st✐♠❛t❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts ✭➓✹✮ ✸ ♣r♦❥❡❝t t❤❡ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ ✭➓✸✳✷✮ ✹ ❞✐♠❡♥s✐♦♥ r❡❞✉❝❡ t♦ ♠❛❦❡ t❤❡ ♣r♦❜❧❡♠ ♠♦r❡ tr❛❝t❛❜❧❡ ✭➓✸✳✹✮ ✺ ❡✈❛❧✉❛t❡ t❤❡ ♣♦rt❢♦❧✐♦ ♣❡r❢♦r♠❛♥❝❡ ❛t t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ ✭➓✺✮

✇❤❛t ✐s ②♦✉r ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥❄

✻ ♣✐❝❦ t❤❡ ♣♦rt❢♦❧✐♦ t❤❛t ♦♣t✐♠✐s❡s ②♦✉r ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✭➓✻✮ ✼ ❛❝❝♦✉♥t ❢♦r ❡st✐♠❛t✐♦♥ r✐s❦ ✶

r❡♣❧❛❝❡ ♣♦✐♥t ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s ✇✐t❤ ❇❛②❡s✐❛♥ ❞✐str✐❜✉t✐♦♥s ✭➓✼✮

✷

r❡✲❡✈❛❧✉❛t❡ t❤❡ ♣♦rt❢♦❧✐♦ ❞✐str✐❜✉t✐♦♥s ✐♥ t❤✐s ❧✐❣❤t ✭➓✽✮

✸

r♦❜✉st❧② r❡✲♦♣t✐♠✐s❡ ✭➓✾✮

❖❜s❡r✈❛t✐♦♥ s❤♦✇s t❤❛t s♦♠❡ st❛t✐st✐❝❛❧ ❢r❡q✉❡♥❝✐❡s ❛r❡✱ ✇✐t❤✐♥ ♥❛rr♦✇❡r ♦r ✇✐❞❡r ❧✐♠✐ts✱ st❛❜❧❡✳ ❇✉t st❛❜❧❡ ❢r❡q✉❡♥❝✐❡s ❛r❡ ♥♦t ✈❡r② ❝♦♠♠♦♥✱ ❛♥❞ ❝❛♥♥♦t ❜❡ ❛ss✉♠❡❞ ❧✐❣❤t❧②✳ ❑❡②♥❡s ✭✶✾✷✶✱ ♣✳✸✽✶✮

✷✵✼ ✴ ✷✷✹

❘❡❣✉❧❛t♦r② ❢r❛♠❡✇♦r❦ ♦❢ r✐s❦ ♠❛♥❛❣❡♠❡♥t

❘❡st✐ ❛♥❞ ❙✐r♦♥✐ ✭✷✵✵✼✱ P❛rt ❱✮ ✐s ❡①t❡♥s✐✈❡

✷✵✽ ✴ ✷✷✹

❆♣♣❡♥❞✐① ❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s ■

❆❝❡r❜✐✱ ❈❛r❧♦ ✭✷✵✵✷✮✳ ✏❙♣❡❝tr❛❧ ♠❡❛s✉r❡s ♦❢ r✐s❦✿ ❛ ❝♦❤❡r❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ s✉❜❥❡❝t✐✈❡ r✐s❦ ❛✈❡rs✐♦♥✑✳ ❏♦✉r♥❛❧ ♦❢ ❇❛♥❦✐♥❣ ❛♥❞ ❋✐♥❛♥❝❡ ✷✻✳✼✱ ♣♣✳ ✶✺✵✺✕✶✺✶✽✳ ❆❝❡r❜✐✱ ❈❛r❧♦ ❛♥❞ ❉✐r❦ ❚❛s❝❤❡ ✭✷✵✵✷✮✳ ✏❖♥ t❤❡ ❝♦❤❡r❡♥❝❡ ♦❢ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧✑✳ ❏♦✉r♥❛❧ ♦❢ ❇❛♥❦✐♥❣ ❛♥❞ ❋✐♥❛♥❝❡ ✷✻✳✼✱ ♣♣✳ ✶✹✽✼✕✶✺✵✸✳ ❆❧❧❡♥✱ ▲✐♥❞❛✱ ❏❛❝♦❜ ❇♦✉❞♦✉❦❤✱ ❛♥❞ ❆♥t❤♦♥② ❙❛✉♥❞❡rs ✭✷✵✵✹✮✳ ❯♥❞❡rst❛♥❞✐♥❣ ♠❛r❦❡t✱ ❝r❡❞✐t✱ ❛♥❞ ♦♣❡r❛t✐♦♥❛❧ r✐s❦✿ t❤❡ ❱❛❧✉❡ ❛t ❘✐s❦ ❛♣♣r♦❛❝❤✳ ❇❧❛❝❦✇❡❧❧ P✉❜❧✐s❤✐♥❣✳ ❆rt③♥❡r✱ P❤✐❧✐♣♣❡ ❡t ❛❧✳ ✭✶✾✾✾✮✳ ✏❈♦❤❡r❡♥t ♠❡❛s✉r❡s ♦❢ r✐s❦✑✳ ▼❛t❤❡♠❛t✐❝❛❧ ❋✐♥❛♥❝❡ ✾✳✸✱ ♣♣✳ ✷✵✸✕✷✷✽✳ ❇❛❦❡r✱ ▼❛❧❝♦❧♠✱ ❇r❡♥❞❛♥ ❇r❛❞❧❡②✱ ❛♥❞ ❏❡✛r❡② ❲✉r❣❧❡r ✭✷✵✶✶✮✳ ✏❇❡♥❝❤♠❛r❦s ❛s ❧✐♠✐ts t♦ ❛r❜✐tr❛❣❡✿ ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ ❧♦✇✲✈♦❧❛t✐❧✐t② ❛♥♦♠❛❧②✑✳ ❋✐♥❛♥❝✐❛❧ ❆♥❛❧②sts ❏♦✉r♥❛❧ ✻✼✳✶✱ ♣♣✳ ✹✵✕✺✹✳

✷✵✾ ✴ ✷✷✹

❆♣♣❡♥❞✐① ❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s ■■

❇❛✉❡r✱ ❘♦❜ ❛♥❞ ❘♦❜✐♥ ❇r❛✉♥ ✭✷✵✶✵✮✳ ✏▼✐s❞❡❡❞s ♠❛tt❡r✿ ❧♦♥❣✲t❡r♠ st♦❝❦ ♣r✐❝❡ ♣❡r❢♦r♠❛♥❝❡ ❛❢t❡r t❤❡ ✜❧✐♥❣ ♦❢ ❝❧❛ss✲❛❝t✐♦♥ ❧❛✇s✉✐ts✑✳ ❋✐♥❛♥❝✐❛❧ ❆♥❛❧②sts ❏♦✉r♥❛❧ ✻✻✳✻✱ ♣♣✳ ✼✹✕✾✷✳ ❇❡r❣✱ ❈❤r✐st✐❛♥ ❛♥❞ ❈❤r✐st♦♣❤❡ ❱✐❣♥❛t ✭✷✵✵✽✮✳ ✏▲✐♥❡❛r✐③❛t✐♦♥ ❝♦❡✣❝✐❡♥ts ♦❢ ❇❡ss❡❧ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❙t✉❞❡♥t t✲❞✐str✐❜✉t✐♦♥s✑✳ ❈♦♥str✉❝t✐✈❡ ❆♣♣r♦①✐♠❛t✐♦♥ ✷✼✳✶✱ ♣♣✳ ✶✺✕✸✷✳ ❇✐❡❧❡❝❦✐✱ ❚♦♠❛s③ ❘✳ ❛♥❞ ▼❛r❡❦ ❘✉t❦♦✇s❦✐ ✭✷✵✵✷✮✳ ❈r❡❞✐t r✐s❦✿ ♠♦❞❡❧✐♥❣✱ ✈❛❧✉❛t✐♦♥ ❛♥❞ ❤❡❞❣✐♥❣✳ ❙♣r✐♥❣❡r ❋✐♥❛♥❝❡✳ ❙♣r✐♥❣❡r✳ ❇♦♦❦st❛❜❡r✱ ❘✐❝❤❛r❞ ✭✷✵✵✼✮✳ ❆ ❉❡♠♦♥ ♦❢ ♦✉r ♦✇♥ ❉❡s✐❣♥✳ ❏♦❤♥ ❲✐❧❡② ❛♥❞ ❙♦♥s✳ ❇♦②❞✱ ❙t❡♣❤❡♥ P✳ ❛♥❞ ▲✐❡✈❡♥ ❱❛♥❞❡♥❜❡r❣❤❡ ✭✷✵✵✹✮✳ ❈♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❇r✐❣♦✱ ❉❛♠✐❛♥♦✱ ▼❛ss✐♠♦ ▼♦r✐♥✐✱ ❛♥❞ ❆♥❞r❡❛ P❛❧❧❛✈✐❝✐♥✐ ✭✷✵✶✸✮✳ ❈♦✉♥t❡r♣❛rt② ❝r❡❞✐t r✐s❦✱ ❝♦❧❧❛t❡r❛❧ ❛♥❞ ❢✉♥❞✐♥❣✳ ❋✐♥❛♥❝❡✳ ❲✐❧❡②✳

✷✶✵ ✴ ✷✷✹

❆♣♣❡♥❞✐① ❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s ■■■

❇r♦❝❦✱ ❲✐❧❧✐❛♠ ❆✳ ❡t ❛❧✳ ✭✶✾✾✻✮✳ ✏❆ ❚❡st ❢♦r ■♥❞❡♣❡♥❞❡♥❝❡ ❇❛s❡❞ ♦♥ t❤❡ ❈♦rr❡❧❛t✐♦♥ ❉✐♠❡♥s✐♦♥✑✳ ❊❝♦♥♦♠❡tr✐❝ ❘❡✈✐❡✇s ✶✺✳✸✱ ♣♣✳ ✶✾✼✕✷✸✺✳ ❈❛♠♣❜❡❧❧✱ ❏♦❤♥ ❨✳✱ ❆♥❞r❡✇ ❲✳ ▲♦✱ ❛♥❞ ❆✳ ❈r❛✐❣ ▼❛❝❑✐♥❧❛② ✭✶✾✾✼✮✳ ❚❤❡ ❊❝♦♥♦♠❡tr✐❝s ♦❢ ❋✐♥❛♥❝✐❛❧ ▼❛r❦❡ts✳ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❈só❦❛✱ Pét❡r✱ P✳ ❏❡❛♥✲❏❛❝q✉❡s ❍❡r✐♥❣s✱ ❛♥❞ ▲ás③❧ó ➪✳ ❑ó❝③② ✭✷✵✵✼✮✳ ✏❈♦❤❡r❡♥t ♠❡❛s✉r❡s ♦❢ r✐s❦ ❢r♦♠ ❛ ❣❡♥❡r❛❧ ❡q✉✐❧✐❜r✐✉♠ ♣❡rs♣❡❝t✐✈❡✑✳ ❏♦✉r♥❛❧ ♦❢ ❇❛♥❦✐♥❣ ❛♥❞ ❋✐♥❛♥❝❡ ✸✶✳✽✱ ♣♣✳ ✷✺✶✼✕✷✺✸✹✳ ❉❛♥✐❡❧ss♦♥✱ ❏♦♥ ❛♥❞ ❏❡❛♥✲P✐❡rr❡ ❩✐❣r❛♥❞ ✭✷✵✵✻✮✳ ✏❖♥ t✐♠❡✲s❝❛❧✐♥❣ ♦❢ r✐s❦ ❛♥❞ t❤❡ sq✉❛r❡✲r♦♦t✲♦❢✲t✐♠❡ r✉❧❡✑✳ ❏♦✉r♥❛❧ ♦❢ ❇❛♥❦✐♥❣ ❛♥❞ ❋✐♥❛♥❝❡ ✸✵✳✶✵✱ ♣♣✳ ✷✼✵✶✕✷✼✶✸✳ ❊❧❧✐s♦♥✱ ▼❛rt✐♥ ❛♥❞ ❚❤♦♠❛s ❏✳ ❙❛r❣❡♥t ✭✷✵✶✷✮✳ ✏❆ ❉❡❢❡♥❝❡ ♦❢ t❤❡ ❋❖▼❈✑✳ ■♥t❡r♥❛t✐♦♥❛❧ ❊❝♦♥♦♠✐❝ ❘❡✈✐❡✇ ✺✸✳✹✱ ♣♣✳ ✶✵✹✼✕✻✺✳ ❊♠❜r❡❝❤ts✱ P❛✉❧ ✭✷✵✵✾✮✳ ✏❈♦♣✉❧❛s✿ ❛ ♣❡rs♦♥❛❧ ✈✐❡✇✑✳ ❏♦✉r♥❛❧ ♦❢ ❘✐s❦ ❛♥❞ ■♥s✉r❛♥❝❡ ✼✻✳✸✱ ♣♣✳ ✻✸✾✕✻✺✵✳

✷✶✶ ✴ ✷✷✹

❆♣♣❡♥❞✐① ❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s ■❱

❊♠❜r❡❝❤ts✱ P❛✉❧✱ ❉♦♠✐♥✐❦ ❉✳ ▲❛♠❜r✐❣❣❡r✱ ❛♥❞ ▼❛r✐♦ ❱✳ ❲üt❤r✐❝❤ ✭✷✵✵✾✮✳ ✏▼✉❧t✐✈❛r✐❛t❡ ❡①tr❡♠❡s ❛♥❞ t❤❡ ❛❣❣r❡❣❛t✐♦♥ ♦❢ ❞❡♣❡♥❞❡♥t r✐s❦s✿ ❡①❛♠♣❧❡s ❛♥❞ ❝♦✉♥t❡r✲❡①❛♠♣❧❡s✑✳ ❊①tr❡♠❡s ✶✷✳✷✱ ♣♣✳ ✶✵✼✕✶✷✼✳ ❊♣st❡✐♥✱ ▲❛rr② ●✳ ❛♥❞ ▼❛rt✐♥ ❙❝❤♥❡✐❞❡r ✭✷✵✶✵✮✳ ✏❆♠❜✐❣✉✐t② ❛♥❞ ❛ss❡t ♠❛r❦❡ts✑✳ ❆♥♥✉❛❧ ❘❡✈✐❡✇ ♦❢ ❋✐♥❛♥❝✐❛❧ ❊❝♦♥♦♠✐❝s ✷✱ ♣♣✳ ✸✶✺✕✸✹✻✳ ❋❛♠❛✱ ❊✉❣❡♥❡ ❋✳ ❛♥❞ ❑❡♥♥❡t❤ ❋r❡♥❝❤ ✭✶✾✾✸✮✳ ✏❈♦♠♠♦♥ ❘✐s❦ ❋❛❝t♦rs ✐♥ t❤❡ ❘❡t✉r♥s ♦♥ ❙t♦❝❦s ❛♥❞ ❇♦♥❞s✑✳ ❏♦✉r♥❛❧ ♦❢ ❋✐♥❛♥❝✐❛❧ ❊❝♦♥♦♠✐❝s ✸✸✳✶✱ ♣♣✳ ✸✕✺✻✳ ❋✐❧✐♣♦✈✐➣✱ ❉❛♠✐r ❛♥❞ ◆✐❝♦❧❛s ❱♦❣❡❧♣♦t❤ ✭✷✵✵✽✮✳ ✏❆ ♥♦t❡ ♦♥ t❤❡ ❙✇✐ss ❙♦❧✈❡♥❝② ❚❡st r✐s❦ ♠❡❛s✉r❡✑✳ ■♥s✉r❛♥❝❡✿ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❊❝♦♥♦♠✐❝s ✹✷✳✸✱ ♣♣✳ ✽✾✼✕✾✵✷✳

• ♦r❞②✱ ▼✐❝❤❛❡❧ ❇✳ ✭✷✵✵✵✮✳ ✏❆ ❝♦♠♣❛r❛t✐✈❡ ❛♥❛t♦♠② ♦❢ ❝r❡❞✐t r✐s❦ ♠♦❞❡❧s✑✳

❏♦✉r♥❛❧ ♦❢ ❇❛♥❦✐♥❣ ❛♥❞ ❋✐♥❛♥❝❡ ✷✹✳✶✕✷✱ ♣♣✳ ✶✶✾✕✶✹✾✳

✷✶✷ ✴ ✷✷✹

❆♣♣❡♥❞✐① ❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s ❱

• ♦✉r✐❡r✱ ❊❧✐s❡✱ ❲❛❧t❡r ❋❛r❦❛s✱ ❛♥❞ ❉♦♥❛t♦ ❆❜❜❛t❡ ✭✷✵✵✾✮✳ ✏❖♣❡r❛t✐♦♥❛❧ r✐s❦

q✉❛♥t✐✜❝❛t✐♦♥ ✉s✐♥❣ ❡①tr❡♠❡ ✈❛❧✉❡ t❤❡♦r② ❛♥❞ ❝♦♣✉❧❛s✿ ❢r♦♠ t❤❡♦r② t♦ ♣r❛❝t✐❝❡✑✳ ❏♦✉r♥❛❧ ♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘✐s❦ ✹✳✸✱ ♣♣✳ ✶✕✷✹✳ ❍❛♥s♦♥✱ ❙❛♠✉❡❧ ●✳✱ ❆♥✐❧ ❑✳ ❑❛s❤②❛♣✱ ❛♥❞ ❏❡r❡♠② ❈✳ ❙t❡✐♥ ✭✷✵✶✶✮✳ ✏❆ ▼❛❝r♦♣r✉❞❡♥t✐❛❧ ❆♣♣r♦❛❝❤ t♦ ❋✐♥❛♥❝✐❛❧ ❘❡❣✉❧❛t✐♦♥✑✳ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝ P❡rs♣❡❝t✐✈❡s ✷✺✳✶✱ ♣♣✳ ✸✕✷✽✳ ❍är❞❧❡✱ ❲♦❧❢❣❛♥❣ ❑❛r❧ ❛♥❞ ❖st❛♣ ❖❦❤r✐♥ ✭✷✵✶✵✮✳ ✏❉❡ ❝♦♣✉❧✐s ♥♦♥ ❡st ❞✐s♣✉t❛♥❞✉♠✿ ❈♦❧✉❧❛❡✿ ❛♥ ♦✈❡r✈✐❡✇✑✳ ❆❙t❆ ❆❞✈❛♥❝❡s ✐♥ ❙t❛t✐st✐❝❛❧ ❆♥❛❧②s✐s ✾✹✳✶✱ ♣♣✳ ✶✕✸✶✳ ❍❡♥♥❡ss②✱ ❉❛✈✐❞ ❆✳ ❛♥❞ ❍❛r✈❡② ❊✳ ▲❛♣❛♥ ✭✷✵✵✻✮✳ ✏❖♥ t❤❡ ♥❛t✉r❡ ♦❢ ❝❡rt❛✐♥t② ❡q✉✐✈❛❧❡♥t ❢✉♥❝t✐♦♥❛❧s✑✳ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❊❝♦♥♦♠✐❝s ✹✸✳✶✳ ❍✉❧❧✱ ❏♦❤♥ ✭✷✵✵✾✮✳ ❖♣t✐♦♥s✱ ❢✉t✉r❡s ❛♥❞ ♦t❤❡r ❞❡r✐✈❛t✐✈❡s✳ ✼t❤✳ P❡❛rs♦♥ Pr❡♥t✐❝❡ ❍❛❧❧✳

✷✶✸ ✴ ✷✷✹

❆♣♣❡♥❞✐① ❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s ❱■

❑❡②♥❡s✱ ❏♦❤♥ ▼❛②♥❛r❞ ✭✶✾✷✶✮✳ ❆ tr❡❛t✐s❡ ♦♥ ♣r♦❜❛❜✐❧✐t②✳ ▲♦♥❞♦♥✿ ▼❛❝▼✐❧❧❛♥ ❛♥❞ ❈♦✳✱ ▲t❞✳ ❑r✐t③♠❛♥✱ ▼❛r❦ ✭✷✵✶✶✮✳ ✏▲♦♥❣ ❧✐✈❡ q✉❛♥t✐t❛t✐✈❡ ♠♦❞❡❧s✦✑ ❈❋❆ ■♥st✐t✉t❡ ▼❛❣❛③✐♥❡✱ ♣♣✳ ✽✕✶✵✳ ▲❛♥❣❡t✐❡❣✱ ❚❡r❡♥❝❡ ❈✳✱ ▼❛rt✐♥ ▲✳ ▲❡✐❜♦✇✐t③✱ ❛♥❞ ❙t❛♥❧❡② ❑♦❣❡❧♠❛♥ ✭✶✾✾✵✮✳ ✏❉✉r❛t✐♦♥ ❚❛r❣❡t✐♥❣ ❛♥❞ t❤❡ ▼❛♥❛❣❡♠❡♥t ♦❢ ▼✉❧t✐♣❡r✐♦❞ ❘❡t✉r♥s✑✳ ❋✐♥❛♥❝✐❛❧ ❆♥❛❧②sts ❏♦✉r♥❛❧ ✹✻✳✺✱ ♣♣✳ ✸✺✕✹✺✳ ▲❡✈②✱ ❍❛✐♠ ✭✶✾✾✷✮✳ ✏❙t♦❝❤❛st✐❝ ❞♦♠✐♥❛♥❝❡ ❛♥❞ ❡①♣❡❝t❡❞ ✉t✐❧✐t②✿ s✉r✈❡② ❛♥❞ ❛♥❛❧②s✐s✑✳ ▼❛♥❛❣❡♠❡♥t ❙❝✐❡♥❝❡ ✸✽✳✹✱ ♣♣✳ ✺✺✺✕✺✾✸✳ ▼❛r❦♦✇✐t③✱ ❍✳ ✭✶✾✺✷✮✳ ✏P♦rt❢♦❧✐♦ s❡❧❡❝t✐♦♥✑✳ ❏♦✉r♥❛❧ ♦❢ ❋✐♥❛♥❝❡ ✼✳✶✱ ♣♣✳ ✼✼✕✾✶✳ ▼❝◆❡✐❧✱ ❆❧❡①❛♥❞❡r ❏✳✱ ❘ü❞✐❣❡r ❋r❡②✱ ❛♥❞ P❛✉❧ ❊♠❜r❡❝❤ts ✭✷✵✶✺✮✳ ◗✉❛♥t✐t❛t✐✈❡ ❘✐s❦ ▼❛♥❛❣❡♠❡♥t✿ ❈♦♥❝❡♣ts✱ ❚❡❝❤♥✐q✉❡s✱ ❛♥❞ ❚♦♦❧s✳ r❡✈✐s❡❞✳ Pr✐♥❝❡t♦♥ ❙❡r✐❡s ✐♥ ❋✐♥❛♥❝❡✳ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✳

✷✶✹ ✴ ✷✷✹

❆♣♣❡♥❞✐① ❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s ❱■■

▼❡✉❝❝✐✱ ❆tt✐❧✐♦ ✭✷✵✵✺✮✳ ❘✐s❦ ❛♥❞ ❆ss❡t ❆❧❧♦❝❛t✐♦♥✳ ❙♣r✐♥❣❡r ❋✐♥❛♥❝❡✳ ❙♣r✐♥❣❡r✳ ✕ ✭✷✵✵✾✮✳ ✏❘❡✈✐❡✇ ♦❢ ❉✐s❝r❡t❡ ❛♥❞ ❈♦♥t✐♥✉♦✉s Pr♦❝❡ss❡s ✐♥ ❋✐♥❛♥❝❡✿ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✑✳ ♠✐♠❡♦✳ ✕ ✭✷✵✶✵✮✳ ✏❚❤❡ ❇❧❛❝❦✲▲✐tt❡r♠❛♥ ❆♣♣r♦❛❝❤✿ ❖r✐❣✐♥❛❧ ▼♦❞❡❧ ❛♥❞ ❊①t❡♥s✐♦♥s✑✳ ♠✐♠❡♦✳ ▼♦r✐♥✐✱ ▼❛ss✐♠♦ ✭✷✵✶✶✮✳ ❯♥❞❡rst❛♥❞✐♥❣ ❛♥❞ ▼❛♥❛❣✐♥❣ ▼♦❞❡❧ ❘✐s❦✿ ❆ Pr❛❝t✐❝❛❧ ●✉✐❞❡ ❢♦r ◗✉❛♥ts✱ ❚r❛❞❡rs ❛♥❞ ❱❛❧✐❞❛t♦rs✳ ❋✐♥❛♥❝❡✳ ❲✐❧❡②✳ ■❙❇◆✿ ✾✼✽✲✵✲✹✼✵✲✾✼✼✻✶✲✸✳ ◆❡❧s❡♥✱ ❘♦❣❡r ❇✳ ✭✷✵✵✻✮✳ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦♣✉❧❛s✳ ✷♥❞✳ ❙♣r✐♥❣❡r✳ ❘❡❜♦♥❛t♦✱ ❘✐❝❝❛r❞♦ ✭✷✵✵✼✮✳ P❧✐❣❤t ♦❢ t❤❡ ❋♦rt✉♥❡ ❚❡❧❧❡rs✿ ❲❤② ❲❡ ◆❡❡❞ t♦ ▼❛♥❛❣❡ ❋✐♥❛♥❝✐❛❧ ❘✐s❦ ❉✐✛❡r❡♥t❧②✳ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❘❡st✐✱ ❆♥❞r❡❛ ❛♥❞ ❆♥❞r❡❛ ❙✐r♦♥✐ ✭✷✵✵✼✮✳ ❘✐s❦ ▼❛♥❛❣❡♠❡♥t ❛♥❞ ❙❤❛r❡❤♦❧❞❡rs✬ ❱❛❧✉❡ ✐♥ ❇❛♥❦✐♥❣✳ ❲✐❧❡② ❋✐♥❛♥❝❡✳ ❏♦❤♥ ❲✐❧❡② ✫ ❙♦♥s✳

✷✶✺ ✴ ✷✷✹

❆♣♣❡♥❞✐① ❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s ❱■■■

❙❝❤r♦❡❞❡r✱ ❆❧✐❝❡ ✭✷✵✵✾✮✳ ❚❤❡ s♥♦✇❜❛❧❧✿ ❲❛rr❡♥ ❇✉✛❡tt ❛♥❞ t❤❡ ❜✉s✐♥❡ss ♦❢ ❧✐❢❡✳ ❇❧♦♦♠s❜✉r② P✉❜❧✐s❤✐♥❣ P▲❈✳ ❙❤❛r♣❡✱ ❲✐❧❧✐❛♠ ❋✳ ✭✶✾✾✶✮✳ ✏❚❤❡ ❆r✐t❤♠❡t✐❝ ♦❢ ❆❝t✐✈❡ ▼❛♥❛❣❡♠❡♥t✑✳ ❋✐♥❛♥❝✐❛❧ ❆♥❛❧②sts✬ ❏♦✉r♥❛❧ ✹✼✳✶✱ ♣♣✳ ✼✕✾✳ ❙❤❡r✐❢✱ ◆❛③♥❡❡♥ ✭✷✵✶✻✮✳ ✏❯❙ ♠♦❞❡❧ r✐s❦ r✉❧❡s ♣✉t ❧✐♦♥s ❜❛❝❦ ✐♥ t❤❡✐r ❝❛❣❡s✑✳ ❘✐s❦✳ ❙❤✐♥✱ ❍②✉♥ ❙♦♥❣ ✭✷✵✶✵✮✳ ❘✐s❦ ❛♥❞ ❧✐q✉✐❞✐t②✳ ❈❧❛r❡♥❞♦♥ ▲❡❝t✉r❡s ✐♥ ❋✐♥❛♥❝❡✳ ❖①❢♦r❞✿ ❖①❢♦r❞ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❙t❡❢❛♥✐❝❛✱ ❉❛♥ ✭✷✵✶✶✮✳ ❆ Pr✐♠❡r ❢♦r t❤❡ ▼❛t❤❡♠❛t✐❝s ♦❢ ❋✐♥❛♥❝✐❛❧ ❊♥❣✐♥❡❡r✐♥❣✳ ✷♥❞✳ ❋✐♥❛♥❝✐❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❆❞✈❛♥❝❡❞ ❇❛❝❦❣r♦✉♥❞ ❙❡r✐❡s✳ ◆❡✇ ❨♦r❦✿ ❋❊ Pr❡ss✳ ❙✉♣❡r✈✐s♦r② ●✉✐❞❛♥❝❡ ♦♥ ♠♦❞❡❧ r✐s❦ ♠❛♥❛❣❡♠❡♥t ✭✷✵✶✶✮✳ ❇✉❧❧❡t✐♥ ❖❈❈ ✷✵✶✶✲✶✷✳ ❇♦❛r❞ ♦❢ ●♦✈❡r♥♦rs ♦❢ t❤❡ ❋❡❞❡r❛❧ ❘❡s❡r✈❡ ❙②st❡♠❀ ❛♥❞ ❖✣❝❡ ♦❢ t❤❡ ❈♦♠♣tr♦❧❧❡r ♦❢ t❤❡ ❈✉rr❡♥❝②✳

✷✶✻ ✴ ✷✷✹

❆♣♣❡♥❞✐① ❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s ■❳

❚❛②❧♦r✱ ❙t❡♣❤❡♥ ❏✳✱ Pr❛❞❡❡♣ ❑✳ ❨❛❞❛✈✱ ❛♥❞ ❨✉❛♥②✉❛♥ ❩❤❛♥❣ ✭✷✵✶✵✮✳ ✏❚❤❡ ✐♥❢♦r♠❛t✐♦♥ ❝♦♥t❡♥t ♦❢ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐❡s ❛♥❞ ♠♦❞❡❧✲❢r❡❡ ✈♦❧❛t✐❧✐t② ❡①♣❡❝t❛t✐♦♥s✿ ❡✈✐❞❡♥❝❡ ❢r♦♠ ♦♣t✐♦♥s ✇r✐tt❡♥ ♦♥ ✐♥❞✐✈✐❞✉❛❧ st♦❝❦s✑✳ ❏♦✉r♥❛❧ ♦❢ ❇❛♥❦✐♥❣ ❛♥❞ ❋✐♥❛♥❝❡ ✸✹✱ ♣♣✳ ✽✼✶✕✽✽✶✳

✷✶✼ ✴ ✷✷✹

❉❡r✐✈❛t✐♦♥s

◆♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ♣❡r❝❡♥t❛❣❡ ❝❤❛♥❣❡s

s✉♣♣♦s❡ t❤❛t Y ∼ N

• µ, σ✷

❛♥❞ X ≡ eY ∼ LogN

• µ, σ✷

t❤❡ ♣❡r❝❡♥t❛❣❡ ❝❤❛♥❣❡ ✐♥ X ✐s t❤❡♥ Xt−Xt−✶

Xt−✶

× ✶✵✵ ❞❡✜♥❡ Z ≡ Xt−Xt−✶

Xt−✶

❢♦r s♠❛❧❧ Z✱ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ②✐❡❧❞s ❧♥ (✶ + Z) = Z − Z ✷ ✷ + · · · ≈ Z t❤✉s✱ ❢♦r s♠❛❧❧ Z✱ eZ ≈ ✶ + Z =

Xt Xt−✶ s♦ Z ≈ ❧♥ Xt − ❧♥ Xt−✶

❛r❡ ✇❡ ❞♦♥❡❄

✷✶✽ ✴ ✷✷✹

❉❡r✐✈❛t✐♦♥s

Pr♦❜❛❜✐❧✐t② ✐♥t❡❣r❛❧ tr❛♥s❢♦r♠

Pr♦♦❢✳

▲❡t U ≡ FX (X) ✇❤❡r❡ FX ✐s ❛♥ ✐♥✈❡rt✐❜❧❡ ❈❉❋❀ t❤✉s✱ U ✐s ❛ r✈✳ ❇② ❞❡✜♥✐t✐♦♥ FU (u) ≡ P {U ≤ u} = P {FX (X) ≤ u} = P {X ≤ QX (u)} = FX (QX (u)) = u. ❚❤✉s✱ U ∼ U ([✵, ✶])✳ ❍♦✇ ✇♦✉❧❞ ②♦✉ ❝♦rr❡❝t t❤❡ ♥❛ï✈❡ st❛t❡♠❡♥t t❤❛t FX (X) = P {X ≤ X}❄

❤✐♥t ✷✶✾ ✴ ✷✷✹

❉❡r✐✈❛t✐♦♥s

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♦❢ ✐s t❛❦❡♥ ❢r♦♠ ▼❡✉❝❝✐ ✭✷✵✵✺✱ ✇✇✇✳✸✳✷✮✿

Pr♦♦❢✳

❇② ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝❢✱ t❤❡ ❝❢ ♦❢ ❳T+τ,˜

τ + ❳T+τ−˜ τ,˜ τ + · · · + ❳T+˜ τ,˜ τ ✐s

φ❳T+τ,˜

τ+❳T+τ−˜ τ,˜ τ+···+❳T+˜ τ,˜ τ (ω) = E

• eiω′(❳T+τ,˜

τ+❳T+τ−˜ τ,˜ τ+···+❳T+˜ τ,˜ τ)

= E

• eiω′❳T+τ,˜

τ × · · · × eiω′❳T+˜ τ,˜ τ

• = E
• eiω′❳T+τ,˜

τ

• × · · · × E
• eiω′❳T+˜

τ,˜ τ

• = φ❳T+τ,˜

τ (ω) × · · · × φ❳T+˜ τ,˜ τ (ω)

=

• φ❳t, ˜

τ (ω)

τ

˜ τ

✇❤❡r❡ t❤❡ ❛♥t❡♣❡♥✉❧t✐♠❛t❡ ❡q✉❛❧✐t② ❝♦♠❡s ❢r♦♠ ✐♥❞❡♣❡♥❞❡♥❝❡✱ ❛♥❞ t❤❡ ✉❧t✐♠❛t❡ ❢r♦♠ ✐❞❡♥t✐❝❛❧✐t②✳

✷✷✵ ✴ ✷✷✹

❉❡r✐✈❛t✐♦♥s

❆❚▼❋ ❧✐q✉✐❞✐t②

❢✉t✉r❡s

st❛♥❞❛r❞✐s❡❞✱ ❡①❝❤❛♥❣❡ tr❛❞❡❞ ❝♦♥tr❛❝ts s❡tt❧❡❞ ❜② ❞❡❧✐✈❡r②

❢♦r✇❛r❞s

❝✉st♦♠✐s❛❜❧❡✱ ❖❚❈ ❝♦♥tr❛❝ts q✉♦t❡❞ ♦♥ P✐♥❦ ◗✉♦t❡✱ ❖❚❈❇❇ ❝❛♥ ❜❡ s❡tt❧❡❞ ✐♥ ❝❛s❤ ✭❙t❡❢❛♥✐❝❛✱ ✷✵✶✶✱ ➓✶✳✶✵✮

❇❛♥❦ ❢♦r ■♥t❡r♥❛t✐♦♥❛❧ ❙❡tt❧❡♠❡♥ts✿ ❖❚❈ ♠❛r❦❡t ♠✉❝❤ ❧❛r❣❡r t❤❛♥ ❡①❝❤❛♥❣❡✲tr❛❞❡❞ ✭❍✉❧❧✱ ✷✵✵✾✱ ♣✳✸✮ ✇❤② ❛r❡ ❆❚▼ ♦♣t✐♦♥s ♠♦st ❧✐q✉✐❞❄

♠♦st ❞❛t❛ ❛r❡ ❆❚▼❀ ❛♥②t❤✐♥❣ ❛✇❛② ❢r♦♠ t❤❛t r❡❧✐❡s ♦♥ ❛ss✉♠♣t✐♦♥s ❢♦❝❛❧ ♣♦✐♥t

✐♠♣ ✈♦❧ ✷✷✶ ✴ ✷✷✹

❉❡r✐✈❛t✐♦♥s

▲❡♠♠❛

E {Ψα} ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s t❤❡ ✉♥✇❡✐❣❤t❡❞ ❛✈❡r❛❣❡ ♦❢ t❤❡ q✉❛♥t✐❧❡s E {Ψα} = ∞

−∞

ψfΨα (ψ) dψ = ✶

✵

QΨα (p) dp

Pr♦♦❢✳

❋♦r ❛♥② ❝♦♥t✐♥✉♦✉s g, g′ ❛♥❞ h✱ ✐♥t❡❣r❛t✐♦♥ ❜② s✉❜st✐t✉t✐♦♥ ❛❧❧♦✇s b

a

h (g (ψ)) g′ (ψ) dψ = g(b)

g(a)

h (p) dp ❚❤✉s✱ ✐❢ a = −∞, b = ∞, g (·) ≡ FΨα (·)✱ ❛♥❞ h (·) ≡ QΨα (·)✿ ∞

−∞

QΨα (FΨα (ψ)) fΨα (ψ) dψ = ✶

✵

QΨα (p) dp ✇❤✐❝❤✱ ❛s QΨα ❛♥❞ FΨα ❛r❡ ♠✉t✉❛❧ ✐♥✈❡rs❡s✱ ❡st❛❜❧✐s❤❡s t❤❡ r❡s✉❧t✳

✷✷✷ ✴ ✷✷✹

❉❡r✐✈❛t✐♦♥s

▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts ✭✷✵✶✺✱ ❊①❛♠♣❧❡ ✷✳✷✺✮

✶ ❢✉❧❧② ❝♦♥❝❡♥tr❛t❡❞ ♣♦rt❢♦❧✐♦

♣♦rt❢♦❧✐♦ ❝♦♥s✐sts ♦❢ ❛ s✐♥❣❧❡ ❞❡❜t ✐♥str✉♠❡♥t ✇✐t❤ ✶✪ ❝❤❛♥❝❡ ♦❢ ❞❡❢❛✉❧t ❞✐str✐❜✉t✐♦♥ ♦❢ Ψα ✐s ❞✐s❝r❡t❡✿ ✳✾✾ ✇❡✐❣❤t ♦♥ ③❡r♦❀ ✳✵✶ ✇❡✐❣❤t ♦♥ ❢✉❧❧ ❧♦ss VaR✾✺ = ✵ ❛s ❞❡❢❛✉❧t ♦❝❝✉rs ✇✐t❤✐♥ t❤❡ t❛✐❧

✷ ❞✐✈❡rs✐✜❡❞ ♣♦rt❢♦❧✐♦

♣♦rt❢♦❧✐♦ ❝♦♥s✐sts ♦❢ ✶✵✵ ✐♥❞❡♣❡♥❞❡♥t ❞❡❜t ✐♥str✉♠❡♥ts✱ ❡❛❝❤ ✇✐t❤ ✶✪ ❝❤❛♥❝❡ ♦❢ ❞❡❢❛✉❧t ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥✿ ♣r♦❜❛❜✐❧✐t② ♦❢ k ♥♦♥✲❞❡❢❛✉❧ts ❢r♦♠ n tr✐❛❧s✱ ❡❛❝❤ ✇✐t❤ s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② p P {X = k} = n k

• pk (✶ − p)n−k

t❤✉s✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♥♦ ❞❡❢❛✉❧ts ✐s ♦♥❧② ✶ − P {X = n} = ✶ − n n

• pn (✶ − p) = ✶ − .✾✾✶✵✵ ≈ .✻✸✹

t❤✉s✱ ❞❡❢❛✉❧ts ♦❝❝✉r ✇❡❧❧ ❜❡❢♦r❡ t❤❡ ❱❛❘ t❤r❡s❤♦❧❞✱ s♦ t❤❛t VaR✾✺ > ✵

✷✷✸ ✴ ✷✷✹

❉❡r✐✈❛t✐♦♥s

❚❤❡ s❡♠✐❞❡✜♥✐t❡ ❝♦♥❡✿ SM

+ ≡ {❙ ✵}

❚❤❡♦r❡♠

SM

+ ✐s ❛ ❝♦♥❡✳

Pr♦♦❢✳

❚❤❡ ♣r♦♦❢ ✉s❡s t❤r❡❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ P❙❉ ♠❛tr✐❝❡s✱ ❙✿ tr (❙) ≥ ✵✱ ③′❙③ ≥ ✵ ❢♦r ❛❧❧ ③ = ✵✱ ❛♥❞ ❛❧❧ ❧❡❛❞✐♥❣ ♣r✐♥❝✐♣❛❧ ♠✐♥♦rs ♦❢ ❙ ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡✳

✶ ✐❢ |❙m| ✐s t❤❡ mt❤ ❧❡❛❞✐♥❣ ♣r✐♥❝✐♣❛❧ ♠✐♥♦r ♦❢ ❙✱ t❤❛t ♦❢ λ❙ ✐s λm |❙m|✳

❆s λ ≥ ✵✱ t❤❡s❡ s❤❛r❡ s✐❣♥s✿ ❙ ❛♥❞ λ❙ t❤✉s s❤❛r❡ s✐❣♥ ❞❡✜♥✐t❡♥❡ss

✷ ❣✐✈❡♥ t✇♦ P❙❉ ♠❛tr✐❝❡s✱ ❙ ❛♥❞ ˜

❙ ③′ ❙ + ˜ ❙

• ③ = ③′❙③ + ③′ ˜

❙③ ≥ ✵.

✸ tr (−❙) = −tr (❙) ≤ ✵✳ ❙❉ ❝♦♥❡ ✷✷✹ ✴ ✷✷✹