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▼♦♥t❡ ❈❛r❧♦ Pr✐❝✐♥❣

❲❤❛t ✇❡ ❤❛✈❡ ❧❡❛r♥❡❞✿

◮ ❆s ❡①❡r❝✐s❡s ✇❡ ❤❛✈❡ ✇r✐tt❡♥ ❛ ❧❛r❣❡ ❛♠♦✉♥t ♦❢ ▼❆❚▲❆❇ st②❧❡

❢✉♥❝t✐♦♥❛❧✐t② ✐♥ t❤❡ ✜❧❡ ♠❛t❧✐❜✳❝♣♣✳ ■♥ ♣❛rt✐❝✉❧❛r ✇❡ ❝❛♥✿

◮ ●❡♥❡r❛t❡ r❛♥❞♦♠ ♥✉♠❜❡rs ✇✐t❤ r❛♥❞❯♥✐❢♦r♠ ❛♥❞ r❛♥❞♥✳ ◮ ●❡♥❡r❛t❡ ♣❧♦ts ✇✐t❤ ♣❧♦t ❛♥❞ ❤✐st✳ ◮ ❈♦♠♣✉t❡ st❛t✐st✐❝s ♦❢ ✈❡❝t♦rs ✇✐t❤ ♠✐♥✱ ♠❡❛♥✱ ♣r❝t✐❧❡ ❡t❝✳

◮ ❲❡ ❤❛✈❡ ❧❡❛r♥❡❞ ❤♦✇ t♦ ✇r✐t❡ s✐♠♣❧❡ ❝❧❛ss❡s✳

❲❤❛t ✇❡ ✇✐❧❧ ❞♦ ♥♦✇✿

◮ ❆❞❞ ❛ ❢✉♥❝t✐♦♥ t♦ ❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧ t♦ ❣❡♥❡r❛t❡ ♣r✐❝❡

♣❛t❤s✳

◮ ❚❡st ♦✉r ♣r✐❝❡ ♣❛t❤s ✉s✐♥❣ ♠❡❛♥ ❡t❝✳ ❛♥❞ ♣❧♦t t❤❡♠ ✉s✐♥❣ ♣❧♦t✳ ◮ ❲r✐t❡ ❛ ❝❧❛ss ▼♦♥t❡❈❛r❧♦Pr✐❝❡r t❤❛t ✉s❡s ❛

❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧ t♦ ❣❡♥❡r❛t❡ ♣r✐❝❡ ♣❛t❤s ❛♥❞ t❤❡♥ ✉s❡s r✐s❦✲♥❡✉tr❛❧ ♣r✐❝✐♥❣ t♦ ♣r✐❝❡ ❛ ❈❛❧❧❖♣t✐♦♥ ❜② ▼♦♥t❡ ❈❛r❧♦✳

• ❡♥❡r❛t❡ ♣r✐❝❡ ♣❛t❤ s♣❡❝✐✜❝❛t✐♦♥

❲❡ ✇✐s❤ t♦ ✇r✐t❡ ❛ ❢✉♥❝t✐♦♥ ❣❡♥❡r❛t❡Pr✐❝❡P❛t❤ ✇❤✐❝❤ t❛❦❡s ❛ ✜♥❛❧ ❞❛t❡ t♦❉❛t❡ ❛♥❞ ❛ ♥✉♠❜❡r ♦❢ st❡♣s ♥❙t❡♣s ❛♥❞ ❣❡♥❡r❛t❡s ❛ r❛♥❞♦♠ ❇❧❛❝❦✕❙❝❤♦❧❡s Pr✐❝❡ ♣❛t❤ ✇✐t❤ t❤❡ ❣✐✈❡♥ ♥✉♠❜❡r ♦❢ st❡♣s✳ ❝❧❛ss✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥④ ♣✉❜❧✐❝✿ ✥✥✥✥✳✳✳✥♦t❤❡r✥♠❡♠❜❡rs✥♦❢✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥✳✳✳ ✥✥✥✥st❞✿✿✈❡❝t♦r❁❞♦✉❜❧❡❃✥❣❡♥❡r❛t❡Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥t♦❉❛t❡✱ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✐♥t✥♥❙t❡♣s✮✥❝♦♥st❀ ⑥❀ ◆♦t❡ t❤❛t t❤❡ ❝❧❛ss ❞❡❝❧❛r❛t✐♦♥ ❡✛❡❝t✐✈❡❧② ❝♦♥t❛✐♥s t❤❡ s♣❡❝✐✜❝❛t✐♦♥✳ ■❢ ②♦✉ ❝❤♦♦s❡ ❣♦♦❞ ❢✉♥❝t✐♦♥ ❛♥❞ ✈❛r✐❛❜❧❡ ♥❛♠❡s✱ ②♦✉ ✇♦♥✬t ♥❡❡❞ t♦♦ ♠❛♥② ❝♦♠♠❡♥ts✳

• ❡♥❡r❛t❡ r✐s❦✲♥❡✉tr❛❧ ♣r✐❝❡ ♣❛t❤ s♣❡❝✐✜❝❛t✐♦♥

❲❡ ❛❧s♦ ✇❛♥t ❛ ❢✉♥❝t✐♦♥ ❣❡♥❡r❛t❡❘✐s❦◆❡✉tr❛❧Pr✐❝❡P❛t❤ ✇❤✐❝❤ ❜❡❤❛✈❡s t❤❡ s❛♠❡✱ ❡①❝❡♣t ✐t ✉s❡s t❤❡ Q✲♠❡❛s✉r❡ t♦ ❝♦♠♣✉t❡ t❤❡ ♣❛t❤✳

❭❜❡❣✐♥④❝♣♣⑥ ❝❧❛ss✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥④ ♣✉❜❧✐❝✿ ✥✥✥✥✳✳✳✥♦t❤❡r✥♠❡♠❜❡rs✥♦❢✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥✳✳✳ ✥✥✥✥st❞✿✿✈❡❝t♦r❁❞♦✉❜❧❡❃✥❣❡♥❡r❛t❡❘✐s❦◆❡✉tr❛❧Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥t♦❉❛t❡✱ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✐♥t✥♥❙t❡♣s✮✥❝♦♥st❀ ⑥❀ ❭❡♥❞④❝♣♣⑥

Pr✐✈❛t❡ ❤❡❧♣❡r ❢✉♥❝t✐♦♥

❚♦ ✐♠♣❧❡♠❡♥t t❤❡s❡ ❢✉♥❝t✐♦♥s✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛ ♣r✐✈❛t❡ ❢✉♥❝t✐♦♥ t❤❛t ❛❧❧♦✇s ②♦✉ t♦ ❝❤♦♦s❡ t❤❡ ❞r✐❢t ✐♥ t❤❡ s✐♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♣r✐❝❡ ♣❛t❤✳

❝❧❛ss✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥④ ✥✥✥✥✳✳✳✥♦t❤❡r✥♠❡♠❜❡rs✥♦❢✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥✳✳✳ ♣r✐✈❛t❡✿ ✥✥✥✥st❞✿✿✈❡❝t♦r❁❞♦✉❜❧❡❃✥❣❡♥❡r❛t❡❘✐s❦◆❡✉tr❛❧Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥t♦❉❛t❡✱ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✐♥t✥♥❙t❡♣s✱ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥❞r✐❢t✮✥❝♦♥st❀ ⑥❀

❚❤✐s ❢✉♥❝t✐♦♥ ✐s ♣r✐✈❛t❡ ❜❡❝❛✉s❡ ✇❡✬✈❡ ♦♥❧② ❝r❡❛t❡❞ ✐t t♦ ♠❛❦❡ t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❡❛s✐❡r✳ ❯s❡rs ♦❢ t❤❡ ❝❧❛ss ❞♦♥✬t ♥❡❡❞ ✭♦r ❡✈❡♥ ✇❛♥t✮ t♦ ❦♥♦✇ ❛❜♦✉t ✐t✳

❆❧❣♦r✐t❤♠ ❢♦r ❇❧❛❝❦✕❙❝❤♦❧❡s ♣r✐❝❡ ♣❛t❤s

❆❧❣♦r✐t❤♠

◮ ❉❡✜♥❡

δti = ti − ti−✶

◮ ❈❤♦♦s❡ ✐♥❞❡♣❡♥❞❡♥t✱ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ǫi✱ ✇✐t❤ ♠❡❛♥ ✵ ❛♥❞

st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ✶✳

◮ ❉❡✜♥❡

sti = sti−✶ +

• µ − ✶

✷σ✷

• δti + σ
• δtiǫi

◮ ❉❡✜♥❡ Sti = ❡①♣(sti)✳ ◮ Sti s✐♠✉❧❛t❡ t❤❡ st♦❝❦ ♣r✐❝❡ ❛t t❤❡ ❞❡s✐r❡❞ t✐♠❡s✳

■♠♣❧❡♠❡♥t t❤❡ ❤❡❧♣❡r ❢✉♥❝t✐♦♥

✈❡❝t♦r❁❞♦✉❜❧❡❃✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✿✿❣❡♥❡r❛t❡Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥t♦❉❛t❡✱ ✥✥✥✥✥✥✥✥✐♥t✥♥❙t❡♣s✱ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥❞r✐❢t✥✮✥❝♦♥st✥④ ✥✥✥✥✈❡❝t♦r❁❞♦✉❜❧❡❃✥♣❛t❤✭♥❙t❡♣s✱✵✳✵✮❀ ✥✥✥✥✈❡❝t♦r❁❞♦✉❜❧❡❃✥❡♣s✐❧♦♥✥❂✥r❛♥❞♥✭✥♥❙t❡♣s✥✮❀ ✥✥✥✥❞♦✉❜❧❡✥❞t✥❂✥✭t♦❉❛t❡✲❞❛t❡✮✴♥❙t❡♣s❀ ✥✥✥✥❞♦✉❜❧❡✥❛✥❂✥✭❞r✐❢t✥✲✥✈♦❧❛t✐❧✐t②✯✈♦❧❛t✐❧✐t②✯✵✳✺✮✯❞t❀ ✥✥✥✥❞♦✉❜❧❡✥❜✥❂✥✈♦❧❛t✐❧✐t②✯sqrt✭❞t✮❀ ✥✥✥✥❞♦✉❜❧❡✥❝✉rr❡♥t▲♦❣❙✥❂✥❧♦❣✭✥st♦❝❦Pr✐❝❡✥✮❀ ✥✥✥✥❢♦r✥✭✐♥t✥✐❂✵❀✥✐❁♥❙t❡♣s❀✥✐✰✰✮✥④ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥❞▲♦❣❙✥❂✥❛✥✰✥❜✯❡♣s✐❧♦♥❬✐❪❀ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥❧♦❣❙✥❂✥❝✉rr❡♥t▲♦❣❙✥✰✥❞▲♦❣❙❀ ✥✥✥✥✥✥✥✥♣❛t❤❬✐❪✥❂✥❡①♣✭✥❧♦❣❙✥✮❀ ✥✥✥✥✥✥✥✥❝✉rr❡♥t▲♦❣❙✥❂✥❧♦❣❙❀ ✥✥✥✥⑥ ✥✥✥✥r❡t✉r♥✥♣❛t❤❀ ⑥

■♠♣❧❡♠❡♥t t❤❡ ♣✉❜❧✐❝ ❢✉♥❝t✐♦♥s

✈❡❝t♦r❁❞♦✉❜❧❡❃✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✿✿❣❡♥❡r❛t❡Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥t♦❉❛t❡✱ ✥✥✥✥✥✥✥✥✐♥t✥♥❙t❡♣s✥✮✥❝♦♥st✥④ ✥✥✥✥r❡t✉r♥✥❣❡♥❡r❛t❡Pr✐❝❡P❛t❤✭✥t♦❉❛t❡✱✥♥❙t❡♣s✱✥❞r✐❢t✥✮❀ ⑥ ✈❡❝t♦r❁❞♦✉❜❧❡❃✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✿✿ ✥✥✥✥❣❡♥❡r❛t❡❘✐s❦◆❡✉tr❛❧Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥t♦❉❛t❡✱ ✥✥✥✥✥✥✥✥✐♥t✥♥❙t❡♣s✥✮✥❝♦♥st✥④ ✥✥✥✥r❡t✉r♥✥❣❡♥❡r❛t❡Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥t♦❉❛t❡✱✥♥❙t❡♣s✱✥r✐s❦❋r❡❡❘❛t❡✥✮❀ ⑥

◆♦t✐❝❡ t❤❛t ✇✐t❤ t❤✐s ❞❡s✐❣♥ ✇❡✬✈❡ ❛✈♦✐❞❡❞ ✇r✐t✐♥❣ t❤❡ s❛♠❡ ❝♦♠♣❧❡① ❝♦❞❡ t✇✐❝❡✳

■♠♣❧❡♠❡♥t ❛ ✈✐s✉❛❧ t❡st

❲❡✬❞ ❧✐❦❡ t♦ s❡❡ ❛ ♣r✐❝❡ ♣❛t❤✱ ✇❡ ❝❛♥ ✉s❡ t❤❡ ▲✐♥❡❈❤❛rt ❝❧❛ss✳

✈♦✐❞✥t❡st❱✐s✉❛❧❧②✭✮✥④ ✥✥✥✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥❜s♠❀ ✥✥✥✥❜s♠✳r✐s❦❋r❡❡❘❛t❡✥❂✥✵✳✵✺❀ ✥✥✥✥❜s♠✳✈♦❧❛t✐❧✐t②✥❂✥✵✳✶❀ ✥✥✥✥❜s♠✳st♦❝❦Pr✐❝❡✥❂✥✶✵✵✳✵❀ ✥✥✥✥❜s♠✳❞❛t❡✥❂✥✷✳✵❀ ✥✥✥✥✐♥t✥♥❙t❡♣s✥❂✥✶✵✵✵❀ ✥✥✥✥❞♦✉❜❧❡✥♠❛t✉r✐t②✥❂✥✹✳✵❀ ✥✥✥✥✈❡❝t♦r❁❞♦✉❜❧❡❃✥♣❛t❤✥❂ ✥✥✥✥✥✥✥❜s♠✳❣❡♥❡r❛t❡Pr✐❝❡P❛t❤✭✥♠❛t✉r✐t②✱✥♥❙t❡♣s✥✮❀ ✥✥✥✥❞♦✉❜❧❡✥❞t✥❂✥✭♠❛t✉r✐t②✲❜s♠✳❞❛t❡✮✴♥❙t❡♣s❀ ✥✥✥✥✈❡❝t♦r❁❞♦✉❜❧❡❃✥t✐♠❡s✥❂ ✥✥✥✥✥✥✥✥❧✐♥s♣❛❝❡✭❞t✱♠❛t✉r✐t②✱♥❙t❡♣s✮❀ ✥✥✥▲✐♥❡❈❤❛rt✥❧✐♥❡❈❤❛rt❀ ✥✥✥❧✐♥❡❈❤❛rt✳s❡t❚✐t❧❡✭✧❙t♦❝❦✥♣r✐❝❡✥♣❛t❤✧✮❀ ✥✥✥❧✐♥❡❈❤❛rt✳s❡t❙❡r✐❡s✭t✐♠❡s✱✥♣❛t❤✮❀ ✥✥✥❧✐♥❡❈❤❛rt✳✇r✐t❡❆s❍❚▼▲✭✧❡①❛♠♣❧❡Pr✐❝❡P❛t❤✳❤t♠❧✧✮❀ ⑥⑥

❊①t❡♥❞✐♥❣ ♠❛t❧✐❜

◮ ❲❡✬✈❡ ✉s❡❞ t❤❡ ❧✐♥s♣❛❝❡ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ♣r❡✈✐♦✉s s❧✐❞❡✳ ◮ ❚❤✐s ✇❛s♥✬t ♦♥❡ ♦❢ t❤❡ ❤♦♠❡✇♦r❦ ❡①❡r❝✐s❡s✱ ❜✉t ✇♦✉❧❞ ❤❛✈❡

❜❡❡♥ ❡❛s② ❡♥♦✉❣❤✳

◮ ❆❞❞✐♥❣ ♥❡✇ ❢✉♥❝t✐♦♥s ❧✐❦❡ t❤✐s t♦ ♠❛t❧✐❜ ✐s s♦ s✐♠♣❧❡ t❤❛t ✇❡

♠❛② ❞♦ s♦ ❢r♦♠ t✐♠❡ t♦ t✐♠❡ ✇✐t❤♦✉t ❜♦t❤❡r✐♥❣ t♦ ♠❡♥t✐♦♥ t❤❛t ✇❡ ❤❛✈❡ ❞♦♥❡ s♦✳

✈♦✐❞✥t❡st❘✐s❦◆❡✉tr❛❧Pr✐❝❡P❛t❤✭✮✥④ ✥✥✥✥r♥❣✭✧❞❡❢❛✉❧t✧✮❀ ✥✥✥✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥❜s♠❀ ✥✥✥✥❜s♠✳r✐s❦❋r❡❡❘❛t❡✥❂✥✵✳✵✺❀ ✥✥✥✥❜s♠✳✈♦❧❛t✐❧✐t②✥❂✥✵✳✶❀ ✥✥✥✥❜s♠✳st♦❝❦Pr✐❝❡✥❂✥✶✵✵✳✵❀ ✥✥✥✥❜s♠✳❞❛t❡✥❂✥✷✳✵❀ ✥✥✥✥✐♥t✥♥P❛t❤s✥❂✥✶✵✵✵✵❀ ✥✥✥✥✐♥t✥♥st❡♣s✥❂✥✺❀ ✥✥✥✥❞♦✉❜❧❡✥♠❛t✉r✐t②✥❂✥✹✳✵❀ ✥✥✥✥✈❡❝t♦r❁❞♦✉❜❧❡❃✥❢✐♥❛❧Pr✐❝❡s✭♥P❛t❤s✱✵✳✵✮❀ ✥✥✥✥❢♦r✥✭✐♥t✥✐❂✵❀✥✐❁♥P❛t❤s❀✥✐✰✰✮✥④ ✥✥✥✥✥✥✥✥✈❡❝t♦r❁❞♦✉❜❧❡❃✥♣❛t❤✥❂ ✥✥✥✥✥✥✥✥✥✥✥✥❜s♠✳❣❡♥❡r❛t❡❘✐s❦◆❡✉tr❛❧Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥♠❛t✉r✐t②✱✥♥st❡♣s✥✮❀ ✥✥✥✥✥✥✥✥❢✐♥❛❧Pr✐❝❡s❬✐❪✥❂✥♣❛t❤✳❜❛❝❦✭✮❀ ✥✥✥✥⑥ ✥✥✥✥❆❙❙❊❘❚❴❆PP❘❖❳❴❊◗❯❆▲✭✥♠❡❛♥✭✥❢✐♥❛❧Pr✐❝❡s✥✮✱ ✥✥✥✥✥✥✥✥❡①♣✭✥❜s♠✳r✐s❦❋r❡❡❘❛t❡✯✷✳✵✮✯❜s♠✳st♦❝❦Pr✐❝❡✱ ✥✥✥✥✥✥✥✥✥✥✥✥✥✵✳✺✮❀ ⑥

❯♥❞❡rst❛♥❞✐♥❣ t❤❡ ❛✉t♦♠❛t❡❞ t❡st

◮ ■❢ ♦✉r r✐s❦✲♥❡✉tr❛❧ ♣r✐❝✐♥❣ ❢✉♥❝t✐♦♥ ✐s ❝♦rr❡❝t✱ t❤❡♥ t❤❡

❞✐s❝♦✉♥t❡❞ ♠❡❛♥ ♦❢ t❤❡ ✜♥❛❧ st♦❝❦ ♣r✐❝❡ s❤♦✉❧❞ ❡q✉❛❧ t❤❡ ✐♥✐t✐❛❧ ♣r✐❝❡✳

◮ ❙✐♥❝❡ t❤✐s t❡st ❞❡♣❡♥❞s ✉♣♦♥ ❣❡♥❡r❛t✐♥❣ r❛♥❞♦♠ ♥✉♠❜❡rs✱ ✇❡

s❡❡❞ t❤❡ r❛♥❞♦♠✲♥✉♠❜❡r ❣❡♥❡r❛t♦r✳ ■✬✈❡ ✇r✐tt❡♥ ❛ ❢✉♥❝t✐♦♥ r♥❣ t♦ ❞♦ t❤✐s✳ ❏✉st ❧✐❦❡ ▼❆❚▲❆❇ ②♦✉ s❤♦✉❧❞ ♣❛ss ✐♥ t❤❡ str✐♥❣ ✬❞❡❢❛✉❧t✬✳

▼♦♥t❡❈❛r❧♦Pr✐❝❡r s♣❡❝✐✜❝❛t✐♦♥

❲❡ ✇❛♥t t♦ ✇r✐t❡ ❛ ❝❧❛ss ❝❛❧❧❡❞ ▼♦♥t❡❈❛r❧♦Pr✐❝❡r t❤❛t✿

◮ ■s ❝♦♥✜❣✉r❡❞ ✇✐t❤ ♥❙❝❡♥❛r✐♦s✱ t❤❡ ♥✉♠❜❡r ♦❢ s❝❡♥❛r✐♦s t♦

❣❡♥❡r❛t❡✳ ❚❤✐s s❤♦✉❧❞ ❞❡❢❛✉❧t t♦ ✶✵✵✵✵✳

◮ ❍❛s ❛ ❢✉♥❝t✐♦♥ ♣r✐❝❡ ✇❤✐❝❤ t❛❦❡s ❛ ❈❛❧❧❖♣t✐♦♥ ❛♥❞ ❛

❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✱ ❛♥❞ ❝♦♠♣✉t❡s ✭❜② ▼♦♥t❡ ❈❛r❧♦✮ t❤❡ ♣r✐❝❡ ♦❢ t❤❡ ❈❛❧❧❖♣t✐♦♥✳ ❲❡✬❧❧ s❡❡ t❤❛t t❤❡ ❞❡❝❧❛r❛t✐♦♥ ❢♦r ▼♦♥t❡❈❛r❧♦Pr✐❝❡r ✐s ♣r❡tt② ♠✉❝❤ t❤❡ s❛♠❡ t❤✐♥❣ ❛s t❤✐s s♣❡❝✐✜❝❛t✐♦♥✳

▼♦♥t❡❈❛r❧♦Pr✐❝❡r ❞❡❝❧❛r❛t✐♦♥

★♣r❛❣♠❛✥♦♥❝❡ ★✐♥❝❧✉❞❡✥✧st❞❛❢①✳❤✧ ★✐♥❝❧✉❞❡✥✧❈❛❧❧❖♣t✐♦♥✳❤✧ ★✐♥❝❧✉❞❡✥✧❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✳❤✧ ❝❧❛ss✥▼♦♥t❡❈❛r❧♦Pr✐❝❡r✥④ ♣✉❜❧✐❝✿ ✥✥✥✥✴✯✥✥❈♦♥str✉❝t♦r✥✯✴ ✥✥✥✥▼♦♥t❡❈❛r❧♦Pr✐❝❡r✭✮❀ ✥✥✥✥✴✯✥✥◆✉♠❜❡r✥♦❢✥s❝❡♥❛r✐♦s✥✯✴ ✥✥✥✥✐♥t✥♥❙❝❡♥❛r✐♦s❀ ✥✥✥✥✴✯✥✥Pr✐❝❡✥❛✥❝❛❧❧✥♦♣t✐♦♥✥✯✴ ✥✥✥✥❞♦✉❜❧❡✥♣r✐❝❡✭✥❝♦♥st✥❈❛❧❧❖♣t✐♦♥✫✥♦♣t✐♦♥✱ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥❝♦♥st✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✫✥♠♦❞❡❧✥✮❀ ⑥❀ ✈♦✐❞✥t❡st▼♦♥t❡❈❛r❧♦Pr✐❝❡r✭✮❀

❘❡✈✐s✐♦♥

◮ ❚❤❡ ❤❡❛❞❡r ✜❧❡ ✐s ❝❛❧❧❡❞ ✳ ✳ ✳ ◮ ❚❤❡ ❤❡❛❞❡r ✜❧❡ ❛❧✇❛②s ❜❡❣✐♥s ✇✐t❤ ✳ ✳ ✳ ◮ ❲❡ ❛❧✇❛②s ★✐♥❝❧✉❞❡ ✳ ✳ ✳ ◮ ❆ ❝♦♥str✉❝t♦r ❧♦♦❦s ❧✐❦❡ ❛ ❢✉♥❝t✐♦♥ ❞❡❝❧❛r❛t✐♦♥ ❡①❝❡♣t ✳ ✳ ✳ ◮ ❲❡ ♣❛ss t❤❡ ♦♣t✐♦♥ ❛♥❞ t❤❡ ♠♦❞❡❧ ❜② ✳ ✳ ✳ ◮ ❲❤❡♥❡✈❡r ✇❡ ✇r✐t❡ ❝♦❞❡ ✇❡ ✳ ✳ ✳ ✐t✳

▼♦♥t❡❈❛r❧♦Pr✐❝❡r✳❝♣♣

★✐♥❝❧✉❞❡✥✧▼♦♥t❡❈❛r❧♦Pr✐❝❡r✳❤✧ ★✐♥❝❧✉❞❡✥✧♠❛t❧✐❜✳❤✧ ✉s✐♥❣✥♥❛♠❡s♣❛❝❡✥st❞❀ ▼♦♥t❡❈❛r❧♦Pr✐❝❡r✿✿▼♦♥t❡❈❛r❧♦Pr✐❝❡r✭✮✥✿ ✥✥✥✥♥❙❝❡♥❛r✐♦s✭✶✵✵✵✵✮✥④ ⑥

❘❡✈✐s✐♦♥

◮ ❚❤❡ ❝♣♣ ✜❧❡ ✐s ❝❛❧❧❡❞ ✳ ✳ ✳ ◮ ❲❡ ❛❧✇❛②s st❛rt ❛ ❝♣♣ ✜❧❡ ✇✐t❤ ✳ ✳ ✳ ◮ ❚❤❡ ❝♦❞❡ ❜❡❣✐♥♥✐♥❣ ▼♦♥t❡❈❛r❧♦Pr✐❝❡r✿✿▼♦♥t❡❈❛r❧♦Pr✐❝❡r

✐s ✳ ✳ ✳

▼♦♥t❡ ❈❛r❧♦ Pr✐❝✐♥❣

❆❧❣♦r✐t❤♠ ✭▼♦♥t❡ ❈❛r❧♦ Pr✐❝✐♥❣✮

❚♦ ❝♦♠♣✉t❡ t❤❡ ❇❧❛❝❦✕❙❝❤♦❧❡s ♣r✐❝❡ ♦❢ ❛♥ ♦♣t✐♦♥ ✇❤♦s❡ ♣❛②♦✛ ✐s ❣✐✈❡♥ ✐♥ t❡r♠s ♦❢ t❤❡ ♣r✐❝❡s ❛t t✐♠❡s t✶✱ t✷✱ ✳ ✳ ✳ ✱ tn✿

◮ ❙✐♠✉❧❛t❡ st♦❝❦ ♣r✐❝❡ ♣❛t❤s ✐♥ t❤❡ r✐s❦✲♥❡✉tr❛❧ ♠❡❛s✉r❡✳ ✐✳❡✳ ✉s❡

t❤❡ ❛❧❣♦r✐t❤♠ ❛❜♦✈❡ ✇✐t❤ µ = r✳

◮ ❈♦♠♣✉t❡ t❤❡ ♣❛②♦✛ ❢♦r ❡❛❝❤ ♣r✐❝❡ ♣❛t❤✳ ◮ ❈♦♠♣✉t❡ t❤❡ ❞✐s❝♦✉♥t❡❞ ♠❡❛♥ ✈❛❧✉❡✳ ◮ ❚❤✐s ❣✐✈❡s ❛♥ ✉♥❜✐❛s❡❞ ❡st✐♠❛t❡ ♦❢ t❤❡ tr✉❡ r✐s❦✲♥❡✉tr❛❧ ♣r✐❝❡✳

❚❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♣r✐❝❡

❞♦✉❜❧❡✥▼♦♥t❡❈❛r❧♦Pr✐❝❡r✿✿♣r✐❝❡✭ ✥✥✥✥✥✥✥✥❝♦♥st✥❈❛❧❧❖♣t✐♦♥✫✥❝❛❧❧❖♣t✐♦♥✱ ✥✥✥✥✥✥✥✥❝♦♥st✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✫✥♠♦❞❡❧✥✮✥④ ✥✥✥✥❞♦✉❜❧❡✥t♦t❛❧✥❂✥✵✳✵❀ ✥✥✥✥❢♦r✥✭✐♥t✥✐❂✵❀✥✐❁♥❙❝❡♥❛r✐♦s❀✥✐✰✰✮✥④ ✥✥✥✥✥✥✥✥✈❡❝t♦r❁❞♦✉❜❧❡❃✥♣❛t❤❂✥♠♦❞❡❧✳ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥❣❡♥❡r❛t❡❘✐s❦◆❡✉tr❛❧Pr✐❝❡P❛t❤✭ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥❝❛❧❧❖♣t✐♦♥✳♠❛t✉r✐t②✱ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✶✥✮❀ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥st♦❝❦Pr✐❝❡✥❂✥♣❛t❤✳❜❛❝❦✭✮❀ ✥✥✥✥✥✥✥✥❞♦✉❜❧❡✥♣❛②♦❢❢✥❂✥❝❛❧❧❖♣t✐♦♥✳♣❛②♦❢❢✭✥st♦❝❦Pr✐❝❡✥✮❀ ✥✥✥✥✥✥✥✥t♦t❛❧✰❂✥♣❛②♦❢❢❀ ✥✥✥✥⑥ ✥✥✥✥❞♦✉❜❧❡✥♠❡❛♥✥❂✥t♦t❛❧✴♥❙❝❡♥❛r✐♦s❀ ✥✥✥✥❞♦✉❜❧❡✥r✥❂✥♠♦❞❡❧✳r✐s❦❋r❡❡❘❛t❡❀ ✥✥✥✥❞♦✉❜❧❡✥❚✥❂✥❝❛❧❧❖♣t✐♦♥✳♠❛t✉r✐t②✥✲✥♠♦❞❡❧✳❞❛t❡❀ ✥✥✥✥r❡t✉r♥✥❡①♣✭✲r✯❚✮✯♠❡❛♥❀ ⑥

❘❡♠❛r❦s

◮ ❲❡ ♦♥❧② ♥❡❡❞ t❤❡ ✜♥❛❧ ♣❛②♦✛ t♦ ♣r✐❝❡ ❛ ❝❛❧❧ ♦♣t✐♦♥✱ s♦ ✇❡ ♦♥❧②

r❡q✉❡st ♦♥❡ st❡♣ ✐♥ t❤❡ ♣r✐❝❡ ♣❛t❤✳

❲❡ ♥❡❡❞ ❛ t❡st

st❛t✐❝✥✈♦✐❞✥t❡stPr✐❝❡❈❛❧❧❖♣t✐♦♥✭✮✥④ ✥✥✥✥r♥❣✭✧❞❡❢❛✉❧t✧✮❀ ✥✥✥✥❈❛❧❧❖♣t✐♦♥✥❝❀ ✥✥✥✥❝✳str✐❦❡✥❂✥✶✶✵❀ ✥✥✥✥❝✳♠❛t✉r✐t②✥❂✥✷❀ ✥✥✥✥❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧✥♠❀ ✥✥✥✥♠✳✈♦❧❛t✐❧✐t②✥❂✥✵✳✶❀ ✥✥✥✥♠✳r✐s❦❋r❡❡❘❛t❡✥❂✥✵✳✵✺❀ ✥✥✥✥♠✳st♦❝❦Pr✐❝❡✥❂✥✶✵✵✳✵❀ ✥✥✥✥♠✳❞r✐❢t✥❂✥✵✳✶❀ ✥✥✥✥♠✳❞❛t❡✥❂✥✶❀ ✥✥✥✥▼♦♥t❡❈❛r❧♦Pr✐❝❡r✥♣r✐❝❡r❀ ✥✥✥✥❞♦✉❜❧❡✥♣r✐❝❡✥❂✥♣r✐❝❡r✳♣r✐❝❡✭✥❝✱✥♠✥✮❀ ✥✥✥✥❞♦✉❜❧❡✥❡①♣❡❝t❡❞✥❂✥❝✳♣r✐❝❡✭✥♠✥✮❀ ✥✥✥✥❆❙❙❊❘❚❴❆PP❘❖❳❴❊◗❯❆▲✭✥♣r✐❝❡✱✥❡①♣❡❝t❡❞✱✥✵✳✶✥✮❀ ⑥

❘❛♥❞♦♠ ◆✉♠❜❡rs

◮ ❚❤❡ ❈✰✰ r❛♥❞♦♠✲♥✉♠❜❡r ❣❡♥❡r❛t♦r r❛♥❞ ✐s♥✬t ✈❡r② ❣♦♦❞✳ ■♥

♣❛rt✐❝✉❧❛r ✐t ✇✐❧❧ ❣✐✈❡ ❜✐❛s❡❞ ❛♥s✇❡rs ❢♦r ❧❛r❣❡ ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥s✳

◮ ❆ st❛♥❞❛r❞ r❛♥❞♦♠✲♥✉♠❜❡r ❣❡♥❡r❛t♦r ❢♦r ▼♦♥t❡ ❈❛r❧♦

s✐♠✉❧❛t✐♦♥s ✐s ❝❛❧❧❡❞ t❤❡ ▼❡rs❡♥♥❡ ❚✇✐st❡r ❛❧❣♦r✐t❤♠✳

◮ ■♠♣❧❡♠❡♥t❡❞ ❛s ♠t✶✾✾✸✼ ✐♥ ❁r❛♥❞♦♠❃ ◮ r❛♥❞❯♥✐❢♦r♠ ❛♥❞ r❛♥❞♥ ❤❛✈❡ ❜❡❡♥ ♠♦❞✐✜❡❞ t♦ ✉s❡ t❤✐s ❝❧❛ss✳

• ❡♥❡r❛t✐♥❣ r❛♥❞♦♠ ♥✉♠❜❡rs

✈❡❝t♦r❁❞♦✉❜❧❡❃✥r❛♥❞✉♥✐❢♦r♠✭✥✐♥t✥♥✥✮✥④ ✥✥✥✥✈❡❝t♦r❁❞♦✉❜❧❡❃✥r❡t✭♥✱✥✵✳✵✮❀ ✥✥✥✥❢♦r✥✭✐♥t✥✐❂✵❀✥✐❁♥❀✥✐✰✰✮✥④ ✥✥✥✥✥✥✥✥r❡t❬✐❪✥❂✥✭♠❡rs❡♥♥❡❚✇✐st❡r✭✮✰✵✳✺✮✴ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥✭♠❡rs❡♥♥❡❚✇✐st❡r✳♠❛①✭✮✰✶✳✵✮❀ ✥✥✥✥⑥ ✥✥✥✥r❡t✉r♥✥r❡t❀ ⑥

◆♦t❡ t❤❡ ✉s❡ ♦❢ ♦♣❡r❛t♦r ♦✈❡r❧♦❛❞✐♥❣✳

❘❡s❡❡❞✐♥❣ t❤❡ r❛♥❞♦♠✲♥✉♠❜❡r ❣❡♥❡r❛t♦r

✈♦✐❞✥r♥❣✭✥❝♦♥st✥str✐♥❣✫✥❞❡s❝r✐♣t✐♦♥✥✮✥④ ✥✥✥✥❆❙❙❊❘❚✭✥❞❡s❝r✐♣t✐♦♥❂❂✧❞❡❢❛✉❧t✧✥✮❀ ✥✥✥✥♠❡rs❡♥♥❡❚✇✐st❡r✳s❡❡❞✭♠t✶✾✾✸✼✿✿❞❡❢❛✉❧t❴s❡❡❞✮❀ ⑥

❲❡ ❛r❡ ✉s✐♥❣ ❛ st❛t✐❝ ✈❛r✐❛❜❧❡ ❤❡r❡✳ ❚❤✐s ✐s ❛ ❣❧♦❜❛❧ ✈❛r✐❛❜❧❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❝❧❛ss✳

❙✉♠♠❛r②

❑❡② ❢✉♥❝t✐♦♥❛❧✐t② ❢♦r t❤❡ ❝♦✉rs❡✿ ♠❛t❧✐❜ ❋✉♥❝t✐♦♥❛❧✐t② s✐♠✐❧❛r t♦ ▼❆❚▲❆❇ ❇❧❛❝❦❙❝❤♦❧❡s▼♦❞❡❧ ❘❡♣r❡s❡♥ts t❤❡ ❇❧❛❝❦✕❙❝❤♦❧❡s ▼♦❞❡❧ ❈❛❧❧❖♣t✐♦♥ ❘❡♣r❡s❡♥ts ❛ ❝❛❧❧ ♦♣t✐♦♥ ❝♦♥tr❛❝t P✉t❖♣t✐♦♥ ❘❡♣r❡s❡♥ts ❛ ♣✉t ♦♣t✐♦♥ ❝♦♥tr❛❝t ▼♦♥t❡❈❛r❧♦Pr✐❝❡r Pr✐❝❡s ♦♣t✐♦♥s ❜② ▼♦♥t❡ ❈❛r❧♦ ◆♦♥ ✜♥❛♥❝✐❛❧ ❢✉♥❝t✐♦♥❛❧✐t②✿ ▲✐♥❡❈❤❛rt P❧♦ts ❧✐♥❡ ❝❤❛rts ❍✐st♦❣r❛♠ P❧♦ts ❤✐st♦❣r❛♠s P✐❡❈❤❛rt P❧♦ts ♣✐❡ ❝❤❛rts ❣❡♦♠❡tr② ❙♦♠❡ ❡❧❡♠❡♥t❛r② ♠❛t❤❡♠❛t✐❝❛❧ ❡①❛♠♣❧❡s ❈♦❞❡ ②♦✉ ❝❛♥ ✉s❡✱ ❜✉t ❞♦♥✬t ❢✉❧❧② ✉♥❞❡rst❛♥❞✿ t❡st✐♥❣ ▼❛❝r♦s t♦ ♠❛❦❡ t❡st✐♥❣ ❧❡ss ❜♦r✐♥❣