t r trs - - PowerPoint PPT Presentation

❙t✉❞②✐♥❣ ❧❛r❣❡ ♥❡t✇♦r❦s ✈✐❛ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t t❤❡♦r②

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠

❊❊❈❙ ❉❡♣❛rt♠❡♥t ❯♥✐✈❡rs✐t② ♦❢ ❈❛❧✐❢♦r♥✐❛✱ ❇❡r❦❡❧❡②

▼❛r❝❤ ✶✵✱ ✷✵✶✼

❆❞✈❛♥❝❡❞ ◆❡t✇♦r❦s ❈♦❧❧♦q✉✐✉♠ ❯♥✐✈❡rs✐t② ♦❢ ▼❛r②❧❛♥❞✱ ❈♦❧❧❡❣❡ P❛r❦

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❏✉st✐♥ ❙❛❧❡③ ❛♥❞ P❛②❛♠ ❉❡❧❣♦s❤❛ ✮

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶ ✴ ✼✵

❖✉t❧✐♥❡

✶

❆ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠ st✉❞✐❡❞ ❜② ❍❛❥❡❦

✷

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❣r❛♣❤s

✸

❚❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❧♦❝❛❧ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡

✹

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❤②♣❡r❣r❛♣❤s

✺

• r❛♣❤ ✐♥❞❡①❡❞ ❞❛t❛

✻

❯♥✐✈❡rs❛❧ ❝♦♠♣r❡ss✐♦♥ ♦❢ ❣r❛♣❤✐❝❛❧ ❞❛t❛

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷ ✴ ✼✵

❏✉st✐♥ ❙❛❧❡③ P❛②❛♠ ❉❡❧❣♦s❤❛

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✸ ✴ ✼✵

❖✉t❧✐♥❡

✶

❆ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠ st✉❞✐❡❞ ❜② ❍❛❥❡❦

✷

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❣r❛♣❤s

✸

❚❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❧♦❝❛❧ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡

✹

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❤②♣❡r❣r❛♣❤s

✺

• r❛♣❤ ✐♥❞❡①❡❞ ❞❛t❛

✻

❯♥✐✈❡rs❛❧ ❝♦♠♣r❡ss✐♦♥ ♦❢ ❣r❛♣❤✐❝❛❧ ❞❛t❛

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹ ✴ ✼✵

❘❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥

❈♦♥s✉♠❡rs ❛❜♦✈❡✱ ❘❡s♦✉r❝❡s ❜❡❧♦✇

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺ ✴ ✼✵

❇❛❧❛♥❝❡❞ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥

▲❡t f ❜❡ ❛ ❝♦♥✈❡① ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ♥♦♥♥❡❣❛t✐✈❡ r❡❛❧s✳ ❖✈❡r ❛ss✐❣♥♠❡♥ts θ✱ t❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ♠✐♥✐♠✐③❡ J(θ) :=

M

• i=✶

f (∂θ(i)) . ✇❤❡r❡ ∂θ(i) ✐s t❤❡ ❧♦❛❞ ❛t r❡s♦✉r❝❡ i ❛♥❞ M ✐s t❤❡ ♥✉♠❜❡r ♦❢ r❡s♦✉r❝❡s✳ ❚❤❡♦r❡♠ ✭ ❍❛❥❡❦✮✿ ❚❤❡ ❛ss✐❣♥♠❡♥t ♠✐♥✐♠✐③❡s ✐✛ ❢♦r ❛❧❧ ♣❛✐rs ♦❢ r❡s♦✉r❝❡s ❛✈❛✐❧❛❜❧❡ t♦ ❝♦♥s✉♠❡r ✇❡ ❤❛✈❡ ✵ ✇❤❡♥❡✈❡r ✳ ◆♦t❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥ ❢♦r ❛♥ ❛ss✐❣♥♠❡♥t t♦ ❜❡ ❜❛❧❛♥❝❡❞ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻ ✴ ✼✵

❇❛❧❛♥❝❡❞ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥

▲❡t f ❜❡ ❛ ❝♦♥✈❡① ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ♥♦♥♥❡❣❛t✐✈❡ r❡❛❧s✳ ❖✈❡r ❛ss✐❣♥♠❡♥ts θ✱ t❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ♠✐♥✐♠✐③❡ J(θ) :=

M

• i=✶

f (∂θ(i)) . ✇❤❡r❡ ∂θ(i) ✐s t❤❡ ❧♦❛❞ ❛t r❡s♦✉r❝❡ i ❛♥❞ M ✐s t❤❡ ♥✉♠❜❡r ♦❢ r❡s♦✉r❝❡s✳ ❚❤❡♦r❡♠ ✭ ❍❛❥❡❦✮✿ ❚❤❡ ❛ss✐❣♥♠❡♥t θ ♠✐♥✐♠✐③❡s J(θ) ✐✛ ❢♦r ❛❧❧ ♣❛✐rs ♦❢ r❡s♦✉r❝❡s i, i′ ❛✈❛✐❧❛❜❧❡ t♦ ❝♦♥s✉♠❡r u ✇❡ ❤❛✈❡ θu(i) = ✵ ✇❤❡♥❡✈❡r ∂θ(i) > ∂θ(i′)✳ ◆♦t❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥ ❢♦r ❛♥ ❛ss✐❣♥♠❡♥t t♦ ❜❡ ❜❛❧❛♥❝❡❞ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ f ✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻ ✴ ✼✵

❯♥✐q✉❡♥❡ss ♦❢ t❤❡ ❜❛❧❛♥❝❡❞ ❧♦❛❞s

❚❤❡ ❛ss✐❣♥♠❡♥t θ ♥❡❡❞ ♥♦t ❜❡ ✉♥✐q✉❡✱ ❜✉t ∂θ(i) ✐s ✉♥✐q✉❡

✶ ✶ ✶ ✶/✷ ✶/✷ ✶/✷ ✶/✷ ✶/✷ ✶/✷

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✼ ✴ ✼✵

▼❛♥② ❝♦♥s✉♠❡rs ❛♥❞ r❡s♦✉r❝❡s

❲❡ ✇❛♥t t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❧♦❝❛❧ ❡♥✈✐r♦♥♠❡♥t ♦❢ ❛ t②♣✐❝❛❧ ❛❣❡♥t ✭❝♦♥s✉♠❡r✱ r❡s♦✉r❝❡✮ ✐♥ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠ ✇✐t❤ ♠❛♥② ❛❣❡♥ts✳ ❲❡ ✇✐❧❧ ✜rst ❞❡s❝r✐❜❡ ❤♦✇ t❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ❢♦r t❤❡ ❜❛s✐❝ ❧♦❛❞ ❜❛❧❛♥❝✐♥❣ ♣r♦❜❧❡♠ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❧❛r❣❡ s♣❛rs❡ ❣r❛♣❤s ✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✽ ✴ ✼✵

❖✉t❧✐♥❡

✶

❆ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠ st✉❞✐❡❞ ❜② ❍❛❥❡❦

✷

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❣r❛♣❤s

✸

❚❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❧♦❝❛❧ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡

✹

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❤②♣❡r❣r❛♣❤s

✺

• r❛♣❤ ✐♥❞❡①❡❞ ❞❛t❛

✻

❯♥✐✈❡rs❛❧ ❝♦♠♣r❡ss✐♦♥ ♦❢ ❣r❛♣❤✐❝❛❧ ❞❛t❛

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✾ ✴ ✼✵

• r❛♣❤s

❆ ❣r❛♣❤ ❝♦rr❡s♣♦♥❞s t♦ ❛ ❧♦❛❞ ❜❛❧❛♥❝✐♥❣ ♣r♦❜❧❡♠ ✇❤❡r❡ ❡❛❝❤ ❝♦♥s✉♠❡r ❤❛s ❛❝❝❡ss t♦ t✇♦ r❡s♦✉r❝❡s✳ ❊❛❝❤ ❡❞❣❡ ✐s ❛ ❝♦♥s✉♠❡r ✇✐t❤ ♦♥❡ ✉♥✐t ♦❢ ❧♦❛❞ ❛♥❞ ❤❛s t♦ ❞❡❝✐❞❡ ❤♦✇ t♦ ❞✐str✐❜✉t❡ ✐ts ❧♦❛❞ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✈❡rt✐❝❡s t❤❛t ❞❡✜♥❡ t❤❡ ❡❞❣❡✳ ▼✉❧t✐♣❧❡ ❡❞❣❡s ❜❡t✇❡❡♥ ❛ ♣❛✐r ♦❢ ✈❡rt✐❝❡s ❛r❡ ♦❦❛②✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✵ ✴ ✼✵

• r❛♣❤s

❆ ❣r❛♣❤ ❝♦rr❡s♣♦♥❞s t♦ ❛ ❧♦❛❞ ❜❛❧❛♥❝✐♥❣ ♣r♦❜❧❡♠ ✇❤❡r❡ ❡❛❝❤ ❝♦♥s✉♠❡r ❤❛s ❛❝❝❡ss t♦ t✇♦ r❡s♦✉r❝❡s✳ ❊❛❝❤ ❡❞❣❡ ✐s ❛ ❝♦♥s✉♠❡r ✇✐t❤ ♦♥❡ ✉♥✐t ♦❢ ❧♦❛❞ ❛♥❞ ❤❛s t♦ ❞❡❝✐❞❡ ❤♦✇ t♦ ❞✐str✐❜✉t❡ ✐ts ❧♦❛❞ ❜❡t✇❡❡♥ t❤❡ t✇♦ ✈❡rt✐❝❡s t❤❛t ❞❡✜♥❡ t❤❡ ❡❞❣❡✳ ▼✉❧t✐♣❧❡ ❡❞❣❡s ❜❡t✇❡❡♥ ❛ ♣❛✐r ♦❢ ✈❡rt✐❝❡s ❛r❡ ♦❦❛②✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✵ ✴ ✼✵

▲♦❛❞ ♣❡r❝♦❧❛t✐♦♥

◆♦t❡ t❤❛t t❤❡ ❧♦❝❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❧♦❜❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❣r❛♣❤✱ ♥♦t ❥✉st ♦♥ ✐ts ❧♦❝❛❧ str✉❝t✉r❡✳

❋✐❣✉r❡✿ ●r❛♣❤ ❆ ❋✐❣✉r❡✿ ●r❛♣❤ ❇

❚❤❡ ♠❛r❦❡❞ ✈❡rt❡① ✐♥ ❣r❛♣❤ ❆ ❤❛s t❤❡ s❛♠❡ ❞❡♣t❤✲✶ ♥❡✐❣❤❜♦r❤♦♦❞ ❛s t❤❡ r♦♦t ✐♥ ❣r❛♣❤ ❇ ✳ ❍♦✇❡✈❡r t❤❡ ✐♥❞✉❝❡❞ ❜❛❧❛♥❝❡❞ ❧♦❛❞ ✐s ✸

✷ ❛t ❡❛❝❤ ✈❡rt❡① ✐♥ ❣r❛♣❤

❆ ❛♥❞ ✐s ✹

✺ ✐♥ ❣r❛♣❤ ❇ ✳

❚❤❡ ♣❤❡♥♦♠❡♥♦♥ ✉♥❞❡r❧②✐♥❣ t❤✐s ✐s ❝❛❧❧❡❞ ❧♦❛❞ ♣❡r❝♦❧❛t✐♦♥ ❜② ❍❛❥❡❦✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✶ ✴ ✼✵

▲♦❛❞ ♣❡r❝♦❧❛t✐♦♥

◆♦t❡ t❤❛t t❤❡ ❧♦❝❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ t❤❡ ❣❧♦❜❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❣r❛♣❤✱ ♥♦t ❥✉st ♦♥ ✐ts ❧♦❝❛❧ str✉❝t✉r❡✳

❋✐❣✉r❡✿ ●r❛♣❤ ❆ ❋✐❣✉r❡✿ ●r❛♣❤ ❇

❚❤❡ ♠❛r❦❡❞ ✈❡rt❡① ✐♥ ❣r❛♣❤ ❆ ❤❛s t❤❡ s❛♠❡ ❞❡♣t❤✲✶ ♥❡✐❣❤❜♦r❤♦♦❞ ❛s t❤❡ r♦♦t ✐♥ ❣r❛♣❤ ❇ ✳ ❍♦✇❡✈❡r t❤❡ ✐♥❞✉❝❡❞ ❜❛❧❛♥❝❡❞ ❧♦❛❞ ✐s ✸

✷ ❛t ❡❛❝❤ ✈❡rt❡① ✐♥ ❣r❛♣❤

❆ ❛♥❞ ✐s ✹

✺ ✐♥ ❣r❛♣❤ ❇ ✳

❚❤❡ ♣❤❡♥♦♠❡♥♦♥ ✉♥❞❡r❧②✐♥❣ t❤✐s ✐s ❝❛❧❧❡❞ ❧♦❛❞ ♣❡r❝♦❧❛t✐♦♥ ❜② ❍❛❥❡❦✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✶ ✴ ✼✵

▲♦❛❞ ♣❡r❝♦❧❛t✐♦♥ ❛s ♥♦♥✉♥✐q✉❡♥❡ss ✐♥ t❤❡ ❧✐♠✐t

❆♥ ✐♥✜♥✐t❡ s♣❛rs❡ ❣r❛♣❤ ❝❛♥ ❡①❤✐❜✐t ♥♦♥✉♥✐q✉❡♥❡ss ✐♥ ✐ts ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥s✳ ■♥ t❤✐s ✐♥✜♥✐t❡ ✸✲r❡❣✉❧❛r tr❡❡✱ st❛rt ❜② ❛ss✐❣♥✐♥❣ t❤❡ ❧♦❛❞ ♦❢ ❡❛❝❤ ❡❞❣❡ t♦ t❤❡ ✈❡rt❡① t❤❛t ✐s ❢✉rt❤❡st ❢r♦♠ t❤❡ ♠❛r❦❡❞ ✈❡rt❡①✳ ❚❤✐s ❣✐✈❡s ✐♥❞✉❝❡❞ ❧♦❛❞ ✶ ❛t ❛❧❧ ✈❡rt✐❝❡s ❡①❝❡♣t ❢♦r t❤❡ ♠❛r❦❡❞ ♦♥❡✱ ✇❤✐❝❤ ❤❛s ✐♥❞✉❝❡❞ ❧♦❛❞ ✵✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✷ ✴ ✼✵

▲♦❛❞ ♣❡r❝♦❧❛t✐♦♥ ❛s ♥♦♥✉♥✐q✉❡♥❡ss ✐♥ t❤❡ ❧✐♠✐t

❆♥ ✐♥✜♥✐t❡ s♣❛rs❡ ❣r❛♣❤ ❝❛♥ ❡①❤✐❜✐t ♥♦♥✉♥✐q✉❡♥❡ss ✐♥ ✐ts ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥s✳ ■♥ t❤✐s ✐♥✜♥✐t❡ ✸✲r❡❣✉❧❛r tr❡❡✱ st❛rt ❜② ❛ss✐❣♥✐♥❣ t❤❡ ❧♦❛❞ ♦❢ ❡❛❝❤ ❡❞❣❡ t♦ t❤❡ ✈❡rt❡① t❤❛t ✐s ❢✉rt❤❡st ❢r♦♠ t❤❡ ♠❛r❦❡❞ ✈❡rt❡①✳ ❚❤✐s ❣✐✈❡s ✐♥❞✉❝❡❞ ❧♦❛❞ ✶ ❛t ❛❧❧ ✈❡rt✐❝❡s ❡①❝❡♣t ❢♦r t❤❡ ♠❛r❦❡❞ ♦♥❡✱ ✇❤✐❝❤ ❤❛s ✐♥❞✉❝❡❞ ❧♦❛❞ ✵✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✷ ✴ ✼✵

◆♦♥✉♥✐q✉❡♥❡ss✿ ❛♥ ❡①❛♠♣❧❡ ❞✉❡ t♦ ❍❛❥❡❦

P✐❝❦ ❛ ♣❛t❤ ❢r♦♠ ✐♥✜♥✐t② t♦ t❤❡ ♠❛r❦❡❞ ♥♦❞❡ ❛♥❞ ✢✐♣ t❤❡ ❛❧❧♦❝❛t✐♦♥s ♦❢ ❡❞❣❡s ❛❧♦♥❣ t❤✐s ♣❛t❤✳ ❚❤✐s ❛❧❧♦❝❛t✐♦♥ ✐s ❜❛❧❛♥❝❡❞✳ ❊❛❝❤ ✈❡rt❡① ❤❛s ✐♥❞✉❝❡❞ ❧♦❛❞ ✶✳ ◆♦✇ ✢✐♣ t❤❡ ❛❧❧♦❝❛t✐♦♥ ♦❢ ❡❛❝❤ ❡❞❣❡✳ ❚❤✐s ✐s ❛♥♦t❤❡r ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ ✦✦ ✳ ❚❤❡ ✐♥❞✉❝❡❞ ❧♦❛❞ ❛t ❡❛❝❤ ✈❡rt❡① ✐s ✷✳ ❚❤❡s❡ ❡①❛♠♣❧❡s ❛r❡ ❞✉❡ t♦ ❍❛❥❡❦✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✸ ✴ ✼✵

◆♦♥✉♥✐q✉❡♥❡ss✿ ❛♥ ❡①❛♠♣❧❡ ❞✉❡ t♦ ❍❛❥❡❦

P✐❝❦ ❛ ♣❛t❤ ❢r♦♠ ✐♥✜♥✐t② t♦ t❤❡ ♠❛r❦❡❞ ♥♦❞❡ ❛♥❞ ✢✐♣ t❤❡ ❛❧❧♦❝❛t✐♦♥s ♦❢ ❡❞❣❡s ❛❧♦♥❣ t❤✐s ♣❛t❤✳ ❚❤✐s ❛❧❧♦❝❛t✐♦♥ ✐s ❜❛❧❛♥❝❡❞✳ ❊❛❝❤ ✈❡rt❡① ❤❛s ✐♥❞✉❝❡❞ ❧♦❛❞ ✶✳ ◆♦✇ ✢✐♣ t❤❡ ❛❧❧♦❝❛t✐♦♥ ♦❢ ❡❛❝❤ ❡❞❣❡✳ ❚❤✐s ✐s ❛♥♦t❤❡r ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ ✦✦ ✳ ❚❤❡ ✐♥❞✉❝❡❞ ❧♦❛❞ ❛t ❡❛❝❤ ✈❡rt❡① ✐s ✷✳ ❚❤❡s❡ ❡①❛♠♣❧❡s ❛r❡ ❞✉❡ t♦ ❍❛❥❡❦✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✸ ✴ ✼✵

❆♥♦t❤❡r ❧♦♦❦ ❛t t❤❡ ❍❛❥❡❦ ❝♦✉♥t❡r❡①❛♠♣❧❡

− →

← −

− → − → ← − − → ← − − → − →

. . . . . . . . .

← − − → ← − ← − − → ← − − → ← − ← −

. . . . . . . . .

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✹ ✴ ✼✵

❍❛❥❡❦✬s ❝♦♥❥❡❝t✉r❡s

❚♦ ❞❡✈❡❧♦♣ ✐♥s✐❣❤t ✐♥t♦ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ❜❛❧❛♥❝❡❞ ❧♦❛❞ ❛❧❧♦❝❛t✐♦♥ ✐♥ ❧❛r❣❡ ❣r❛♣❤s ❍❛❥❡❦ ❝❛rr✐❡❞ ♦✉t s✐♠✉❧❛t✐♦♥s✳ ❍❡ ♣✐❝❦❡❞ r❛♥❞♦♠ ❣r❛♣❤s ❛❝❝♦r❞✐♥❣ t♦ ❛ s♣❛rs❡ ❊r❞➤s✲❘é♥②✐ ♠♦❞❡❧ ❛♥❞ st✉❞✐❡❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥s✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✺ ✴ ✼✵

❆ s♣❛rs❡ ❊r❞➤s✲❘é♥②✐ ❣r❛♣❤

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✻ ✴ ✼✵

◆✉♠❡r✐❝s ♦♥ ❊r❞➤s✲❘é♥②✐ ❣r❛♣❤s ✭❍❛❥❡❦✮

αM ❝♦♥s✉♠❡rs ❛♥❞ M r❡s♦✉r❝❡s❀ ❡❞❣❡s ♣✐❝❦❡❞ ❛t r❛♥❞♦♠

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✼ ✴ ✼✵

◆✉♠❡r✐❝s ♦♥ ❊r❞➤s✲❘é♥②✐ ❣r❛♣❤s ✭❍❛❥❡❦✮ ✭❝♦♥t✬❞✮

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✽ ✴ ✼✵

▲❛r❣❡ ❊r❞➤s ❘é♥②✐ ❣r❛♣❤s

G(n, α/n)

✶ ❇❡r P♦✐

✸

✷ ✷

✶

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✾ ✴ ✼✵

▲❛r❣❡ ❊r❞➤s ❘é♥②✐ ❣r❛♣❤s

G(n, α/n)

(n − ✶)❇❡r(α/n) ≈ P♦✐(α)

✸

✷ ✷

✶

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✾ ✴ ✼✵

▲❛r❣❡ ❊r❞➤s ❘é♥②✐ ❣r❛♣❤s

G(n, α/n)

(n − ✶)❇❡r(α/n) ≈ P♦✐(α)

✸

✷ ✷

✶

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✾ ✴ ✼✵

▲❛r❣❡ ❊r❞➤s ❘é♥②✐ ❣r❛♣❤s

G(n, α/n)

(n − ✶)❇❡r(α/n) ≈ P♦✐(α)

(n − ✸) α✷

n✷ = O(✶/n) ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✶✾ ✴ ✼✵

❚❤❡ P♦✐ss♦♥ ●❛❧t♦♥✲❲❛ts♦♥ tr❡❡

P♦✐ss♦♥ ●❛❧t♦♥✲❲❛ts♦♥ tr❡❡ ✿ ✕ ❙t❛rt ✇✐t❤ ❛ r♦♦t✳ ✕ P✐❝❦ ❛ P♦✐ss♦♥ ✭λ✮ ♥✉♠❜❡r ♦❢ ♥❡✐❣❤❜♦rs ✭❛t ❞❡♣t❤ ✶✮✳ ✕ ❋♦r ❡❛❝❤ ♦❢ t❤❡s❡✱ ✐♥❞❡♣❡♥❞❡♥t❧② ♣✐❝❦ ❛ P♦✐ss♦♥ ✭λ✮ ♥✉♠❜❡r ♦❢ ♥❡✐❣❤❜♦rs ✭❛t ❞❡♣t❤ ✷✮✳ ✳ ✳ ✳ ❊t❝✳ ❚❤❡ ❧♦❝❛❧ ❡♥✈✐r♦♥♠❡♥t ♦❢ ❛ t②♣✐❝❛❧ ✈❡rt❡① ✐♥ ❛♥ ❊r❞➤s ✲ ❘é♥②✐ ❣r❛♣❤ ❝♦♥✈❡r❣❡s t♦ ❛ P♦✐ss♦♥ ●❛❧t♦♥✲❲❛ts♦♥ tr❡❡ ❛s ✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✵ ✴ ✼✵

❚❤❡ P♦✐ss♦♥ ●❛❧t♦♥✲❲❛ts♦♥ tr❡❡

P♦✐ss♦♥ ●❛❧t♦♥✲❲❛ts♦♥ tr❡❡ ✿ ✕ ❙t❛rt ✇✐t❤ ❛ r♦♦t✳ ✕ P✐❝❦ ❛ P♦✐ss♦♥ ✭λ✮ ♥✉♠❜❡r ♦❢ ♥❡✐❣❤❜♦rs ✭❛t ❞❡♣t❤ ✶✮✳ ✕ ❋♦r ❡❛❝❤ ♦❢ t❤❡s❡✱ ✐♥❞❡♣❡♥❞❡♥t❧② ♣✐❝❦ ❛ P♦✐ss♦♥ ✭λ✮ ♥✉♠❜❡r ♦❢ ♥❡✐❣❤❜♦rs ✭❛t ❞❡♣t❤ ✷✮✳ ✳ ✳ ✳ ❊t❝✳ ❚❤❡ ❧♦❝❛❧ ❡♥✈✐r♦♥♠❡♥t ♦❢ ❛ t②♣✐❝❛❧ ✈❡rt❡① ✐♥ ❛♥ ❊r❞➤s ✲ ❘é♥②✐ ❣r❛♣❤ ❝♦♥✈❡r❣❡s t♦ ❛ P♦✐ss♦♥ ●❛❧t♦♥✲❲❛ts♦♥ tr❡❡ ❛s M → ∞✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✵ ✴ ✼✵

❆ r❡❝✉rs✐✈❡ ❞✐str✐❜✉t✐♦♥❛❧ ❡q✉❛t✐♦♥

❚❤❡ ♥✉♠❡r✐❝s s✉❣❣❡st t❤❛t t❤❡r❡ s❤♦✉❧❞ ❜❡ ❛ ✇❡❧❧ ❞❡✜♥❡❞ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ✭M → ∞✮ ❢♦r t❤❡ ✐♥❞✉❝❡❞ ❧♦❛❞ ✭✐♥ ❛ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥✮ ❛t ❛ t②♣✐❝❛❧ ✈❡rt❡①✳ ◆❛t✉r❛❧ ❣✉❡ss✿ t❤❡ ❧✐♠✐t✐♥❣ ✐♥❞✉❝❡❞ ❧♦❛❞ ❞✐str✐❜✉t✐♦♥ ♦❜❡②s ❛ ✜①❡❞ ♣♦✐♥t ❡q✉❛t✐♦♥ ✭❛ r❡❝✉rs✐✈❡ ❞✐str✐❜✉t✐♦♥❛❧ ❡q✉❛t✐♦♥ ✮✳ ❚❤✐s ✇❛s ❝♦♥❥❡❝t✉r❡❞ ❜② ❍❛❥❡❦✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✶ ✴ ✼✵

❖✉r ❝♦♥tr✐❜✉t✐♦♥

❲❡ ✈❡r✐❢② t❤✐s ❝♦♥❥❡❝t✉r❡ ♦❢ ❍❛❥❡❦ ❛s ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❛ ❜r♦❛❞❡r r❡s✉❧t✳ ❖✉r r❡s✉❧ts ❛r❡ ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ❧♦❝❛❧ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❡q✉❡♥❝❡s ♦❢ ❣r❛♣❤s✱ ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ♦❜❥❡❝t✐✈❡ ♠❡t❤♦❞ ✳ ■♥ t❤✐s t❤❡♦r② ❣r❛♣❤s ❛r❡ ✈✐❡✇❡❞ t❤r♦✉❣❤ t❤❡ ❧❡♥s ♦❢ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s ♦♥ r♦♦t❡❞ ❣r❛♣❤s✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✷ ✴ ✼✵

❲❤❛t ✇❡ ♣r♦✈❡ ✭✇✐t❤ ❏✉st✐♥ ❙❛❧❡③✮

❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡❧② ❞❡✜♥❡❞ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ ❛ss♦❝✐❛t❡❞ t♦ ❛♥② ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ♦♥ ✐♥✜♥✐t❡ r♦♦t❡❞ ❣r❛♣❤s t❤❛t ❝❛♥ ❛r✐s❡ ❛s ❛ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ ❣r❛♣❤s✳ ❚❤❡ ✉♥✐q✉❡ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ ♦♥ t❤❡ ✜♥✐t❡ ❣r❛♣❤s ❝♦♥✈❡r❣❡s t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✉♥✐q✉❡ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ ♦♥ ✐ts ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t✳ ❚❤❡ ✐♥❞✉❝❡❞ ❧♦❛❞ ❞✐str✐❜✉t✐♦♥ ❛t t❤❡ r♦♦t ✐♥ t❤❡ ✐♥✜♥✐t❡ ❧✐♠✐t r♦♦t❡❞ ❣r❛♣❤ ♦❜❡②s t❤❡ ❡①♣❡❝t❡❞ r❡❝✉rs✐✈❡ ❞✐str✐❜✉t✐♦♥❛❧ ❡q✉❛t✐♦♥✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✸ ✴ ✼✵

❖✉t❧✐♥❡

✶

❆ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠ st✉❞✐❡❞ ❜② ❍❛❥❡❦

✷

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❣r❛♣❤s

✸

❚❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❧♦❝❛❧ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡

✹

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❤②♣❡r❣r❛♣❤s

✺

• r❛♣❤ ✐♥❞❡①❡❞ ❞❛t❛

✻

❯♥✐✈❡rs❛❧ ❝♦♠♣r❡ss✐♦♥ ♦❢ ❣r❛♣❤✐❝❛❧ ❞❛t❛

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✹ ✴ ✼✵

❙t♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❛s ❛ ♠♦❞❡❧ ❢♦r ❞❛t❛ s❛♠♣❧❡s

❆ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✐s ❛ ♠♦❞❡❧ ❢♦r t❤❡ str✉❝t✉r❡ ♦❢ ❞❛t❛ s❛♠♣❧❡s✳

✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶ ✶ ✶

✶ ✷ ✶

✶

✶ ✵ ✶ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✺ ✴ ✼✵

❙t♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❛s ❛ ♠♦❞❡❧ ❢♦r ❞❛t❛ s❛♠♣❧❡s

❆ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✐s ❛ ♠♦❞❡❧ ❢♦r t❤❡ str✉❝t✉r❡ ♦❢ ❞❛t❛ s❛♠♣❧❡s✳ −N N

✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶ ✶ ✶

L ✶ ✷ ✶

✶

✶ ✵ ✶ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✺ ✴ ✼✵

❙t♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❛s ❛ ♠♦❞❡❧ ❢♦r ❞❛t❛ s❛♠♣❧❡s

❆ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✐s ❛ ♠♦❞❡❧ ❢♦r t❤❡ str✉❝t✉r❡ ♦❢ ❞❛t❛ s❛♠♣❧❡s✳ −N N

✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶ ✶ ✶

L ✶ ✷(N + ✶) − L

N−L+✶

• i=−N

δxi,...,xi+L−✶ ⇒ PX✵,...,XL−✶.

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✺ ✴ ✼✵

✏❊♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥✑ ♦❢ ❛ ♠❛r❦❡❞ ❣r❛♣❤

✹ ✷ ✸ ✶ ✺ ✻ ✼ ✽

G

✶ ✹ ✶ ✷ ✶ ✹

U✷(G)

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✻ ✴ ✼✵

❘♦♦t❡❞ ♠❛r❦❡❞ ❣r❛♣❤ ♣r♦❝❡ss ❢r♦♠ ❛ ♠❛r❦❡❞ ❣r❛♣❤

✹ ✷ ✸ ✶ ✺ ✻ ✼ ✽

G

✶ ✹ ✶ ✷ ✶ ✹

U(G)

✿ s♣❛❝❡ ♦❢ ✉♥❧❛❜❡❧❧❡❞ ♠❛r❦❡❞ r♦♦t❡❞ ❣r❛♣❤s ❆ ♣r♦❝❡ss ✇✐t❤ ✈❛❧✉❡s ✐♥ r♦♦t❡❞ ♠❛r❦❡❞ ❣r❛♣❤s✿ ❲❡ ✇✐❧❧ ✜rst ❝♦♥s✐❞❡r t❤❡ ✉♥♠❛r❦❡❞ ❝❛s❡✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✼ ✴ ✼✵

❘♦♦t❡❞ ♠❛r❦❡❞ ❣r❛♣❤ ♣r♦❝❡ss ❢r♦♠ ❛ ♠❛r❦❡❞ ❣r❛♣❤

✹ ✷ ✸ ✶ ✺ ✻ ✼ ✽

G

✶ ✹ ✶ ✷ ✶ ✹

U(G)

G∗✿ s♣❛❝❡ ♦❢ ✉♥❧❛❜❡❧❧❡❞ ♠❛r❦❡❞ r♦♦t❡❞ ❣r❛♣❤s ❆ ♣r♦❝❡ss ✇✐t❤ ✈❛❧✉❡s ✐♥ r♦♦t❡❞ ♠❛r❦❡❞ ❣r❛♣❤s✿ ❲❡ ✇✐❧❧ ✜rst ❝♦♥s✐❞❡r t❤❡ ✉♥♠❛r❦❡❞ ❝❛s❡✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✼ ✴ ✼✵

❘♦♦t❡❞ ♠❛r❦❡❞ ❣r❛♣❤ ♣r♦❝❡ss ❢r♦♠ ❛ ♠❛r❦❡❞ ❣r❛♣❤

✹ ✷ ✸ ✶ ✺ ✻ ✼ ✽

G

✶ ✹ ✶ ✷ ✶ ✹

U(G)

G∗✿ s♣❛❝❡ ♦❢ ✉♥❧❛❜❡❧❧❡❞ ♠❛r❦❡❞ r♦♦t❡❞ ❣r❛♣❤s ❆ ♣r♦❝❡ss ✇✐t❤ ✈❛❧✉❡s ✐♥ r♦♦t❡❞ ♠❛r❦❡❞ ❣r❛♣❤s✿ µ ∈ P(G∗) ❲❡ ✇✐❧❧ ✜rst ❝♦♥s✐❞❡r t❤❡ ✉♥♠❛r❦❡❞ ❝❛s❡✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✼ ✴ ✼✵

❚❤❡ s♣❛❝❡ ♦❢ r♦♦t❡❞ ❣r❛♣❤s

G∗ ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❧♦❝❛❧❧② ✜♥✐t❡ ❝♦♥♥❡❝t❡❞ r♦♦t❡❞ ❣r❛♣❤s ❝♦♥s✐❞❡r❡❞ ✉♣ t♦ r♦♦t❡❞ ✐s♦♠♦r♣❤✐s♠✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❡❧❡♠❡♥ts ♦❢ G∗ ✐s

✶ ✶+r ✱ ✇❤❡r❡ r ✐s t❤❡

❧❛r❣❡st ❞❡♣t❤ ♦❢ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ❛r♦✉♥❞ t❤❡ r♦♦t ✉♣ t♦ ✇❤✐❝❤ t❤❡② ❛❣r❡❡✳ ❚❤✐s ❞✐st❛♥❝❡ ♠❛❦❡s G∗ ✐♥t♦ ❛ ❝♦♠♣❧❡t❡ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✽ ✴ ✼✵

▲♦❝❛❧ ✇❡❛❦ ❧✐♠✐t ♦❢ ❛ s❡q✉❡♥❝❡ ♦❢ ❣r❛♣❤s

❆ ✜①❡❞ ✜♥✐t❡ ❣r❛♣❤ G ❝♦rr❡s♣♦♥❞s t♦ ❛ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ♦♥ G∗ ❜② ♣✐❝❦✐♥❣ t❤❡ r♦♦t ❛t r❛♥❞♦♠ ❢r♦♠ t❤❡ ✈❡rt✐❝❡s ♦❢ G✳ ❆ s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ ❣r❛♣❤s ✐s s❛✐❞ t♦ ❝♦♥✈❡r❣❡ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❧♦❝❛❧ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ✐❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s ♦♥ G∗ ❝♦♥✈❡r❣❡ ✇❡❛❦❧②✳ ❚❤❡ ❞❡✜♥✐t✐♦♥s ❡①t❡♥❞ ♥❛t✉r❛❧❧② t♦ ♠❛r❦❡❞ ❣r❛♣❤s ✱ ✐✳❡✳ ❣r❛♣❤s ✇❤❡r❡ ❡❛❝❤ ❡❞❣❡ ❛♥❞ ❡❛❝❤ ✈❡rt❡① ❝❛rr✐❡s ❛♥ ❡❧❡♠❡♥t ♦❢ s♦♠❡ ♦t❤❡r s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✾ ✴ ✼✵

❚❤❡ s♣❛❝❡ ♦❢ ✭❡❞❣❡✱ ✈❡rt❡①✮ r♦♦t❡❞ ❣r❛♣❤s

G∗∗ ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ❧♦❝❛❧❧② ✜♥✐t❡ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤s ✇✐t❤ ❛ ❞✐st✐♥❣✉✐s❤❡❞ ♦r✐❡♥t❡❞ ❡❞❣❡✱ ❝♦♥s✐❞❡r❡❞ ✉♣ t♦ ✐s♦♠♦r♣❤✐s♠ ✭♣r❡s❡r✈✐♥❣ t❤❡ ❞✐st✐♥❣✉✐s❤❡❞ ♦r✐❡♥t❡❞ ❡❞❣❡✮✳ G∗∗ ❝❛♥ ❜❡ ♠❡tr✐③❡❞ t♦ ❣✐✈❡ ❛ ❝♦♠♣❧❡t❡ s❡♣❛r❛❜❧❡ ♠❡tr✐❝ s♣❛❝❡✱ ❥✉st ❛s ❢♦r G∗✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✸✵ ✴ ✼✵

▼♦✈✐♥❣ ❜❡t✇❡❡♥ G∗ ❛♥❞ G∗∗

❆ ❢✉♥❝t✐♦♥ f : G∗∗ → R ❣✐✈❡s r✐s❡ t♦ ❛ ❢✉♥❝t✐♦♥ ∂f : G∗ → R ✈✐❛ ∂f (G, o) =

• i∼o

f (G, i, o) . ❆ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ µ ♦♥ G∗ ❣✐✈❡s r✐s❡ t♦ ❛ ♠❡❛s✉r❡ µ ♦♥ G∗∗ ✈✐❛

• G∗∗

fd µ =

• G∗

∂fdµ , ❢♦r ❛❧❧ ❜♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s f . ◆♦t❡ t❤❛t µ(G∗∗) = ❞❡❣(µ) :=

• G∗ ❞❡❣(r♦♦t)dµ ✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✸✶ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r✐t②

• ✐✈❡♥ f

: G∗∗ → R✱ ❞❡✜♥❡ f ∗ : G∗∗ → R ✈✐❛ f ∗(G, i, o) = f (G, o, i) . ❆ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ µ ♦♥ G∗ ✐s ❝❛❧❧❡❞ ✉♥✐♠♦❞✉❧❛r ✐❢

• G∗∗

fd µ =

• G∗∗

f ∗d µ , ❢♦r ❛❧❧ ❜♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s f . ■t ✐s ❦♥♦✇♥ t❤❛t t❤❡ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t ♦❢ ❛♥② s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ ❣r❛♣❤s ✐s ✉♥✐♠♦❞✉❧❛r ✭❆❧❞♦✉s ❛♥❞ ▲②♦♥s✮✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✸✷ ✴ ✼✵

❆s②♠♣t♦t✐❝ ♥♦t✐♦♥ ♦❢ ❛ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥

❆ ❢✉♥❝t✐♦♥ Θ : G∗∗ → [✵, ✶] ✐s ❝❛❧❧❡❞ ❛♥ ❛❧❧♦❝❛t✐♦♥ ✐❢ Θ + Θ∗ = ✶✳ ❆♥ ❛❧❧♦❝❛t✐♦♥ Θ ✐s ❝❛❧❧❡❞ ❛ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ ❢♦r ❛ ❣✐✈❡♥ ✉♥✐♠♦❞✉❧❛r µ ✐❢ ❢♦r µ ❛❧♠♦st ❛❧❧ (G, i.o) ✐t ❤♦❧❞s t❤❛t ∂Θ(G, i) < ∂Θ(G, o) = ⇒ Θ(G, i, o) = ✵ .

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✸✸ ✴ ✼✵

❋♦r♠❛❧ st❛t❡♠❡♥t ♦❢ t❤❡ ♠❛✐♥ r❡s✉❧ts

❲❡ ♣r♦✈❡ t❤❛t ❢♦r ❛♥② ✉♥✐♠♦❞✉❧❛r µ ✇✐t❤ ❞❡❣(µ) < ∞ t❤❡r❡ ✐s ❛ Θ✵ t❤❛t ✐s ❛ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ ❢♦r µ ✇✐t❤ t❤❡ ♣r♦♣❡rt② t❤❛t ✐t s✐♠✉❧t❛♥❡♦✉s❧② ♠✐♥✐♠✐③❡s

• G∗ f (∂Θ)dµ ♦✈❡r ❛❧❧♦❝❛t✐♦♥s Θ ❢♦r

❡✈❡r② ❝♦♥✈❡① r❡❛❧ ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ f ♦♥ R+✳ ❋✉rt❤❡r✱ Θ✵ ✐s µ✲❛❧♠♦st s✉r❡❧② ✉♥✐q✉❡✳ ❋♦r ❛♥② s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡ ❣r❛♣❤s ✇✐t❤ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t µ✱ t❤❡ ❡♠♣✐r✐❝✐❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♥❞✉❝❡❞ ❧♦❛❞ ✐♥ t❤❡ ✉♥✐q✉❡ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ ♦♥ t❤❡s❡ ❣r❛♣❤s ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ t❤❡ ❧❛✇ ♦❢ ∂Θ✵ ✭❢♦r t❤❡ Θ✵ ♦❢ t❤❡ ❧✐♠✐t✮✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✸✹ ✴ ✼✵

❱❛r✐❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t

• ✐✈❡♥ ✉♥✐♠♦❞✉❧❛r µ ♦♥ G∗ ✇✐t❤ ❞❡❣(µ) < ∞✱ ❞❡✜♥❡✱ ❢♦r ❡❛❝❤

t ≥ ✵✱ Φµ(t) :=

• G∗

(∂Θ✵ − t)+dµ . t → Φµ(t) ✐s t❤❡ ♠❡❛♥✲❡①❝❡ss ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❛❧♠♦st s✉r❡❧② ✉♥✐q✉❡ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ ❛ss♦❝✐❛t❡❞ t♦ µ✳ ❲❡ ❤❛✈❡ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ Φµ(t) = ♠❛①

f : G∗→[✵,✶],❇♦r❡❧

{✶ ✷

• G∗∗

ˆ f d µ − t

• G∗

fdµ} , ❢♦r ❡❛❝❤ t✱ ✇❤❡r❡ ˆ f (G, i, o) := f (G, i) ∧ f (G, o) .

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✸✺ ✴ ✼✵

■♥t✉✐t✐♦♥ ❜❡❤✐♥❞ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥

❚❤❡ ♦♣t✐♠✐③✐♥❣ ❢✉♥❝t✐♦♥ ✐s f = ✶(∂Θ✵ > t)✳ ❚♦ ❝❤❡❝❦ t❤✐s✱ ♦❜s❡r✈❡ t❤❛t ✶ ✷

• G∗∗

ˆ f d µ = ✶ ✷

• G∗

(∂ ˆ f )dµ = ✶ ✷

• G∗
• i∼o

✶(∂Θ✵(G, i) > t ❛♥❞ ∂Θ✵(G, o) > t)dµ ❚❤✉s

• G∗

(∂Θ✵ − t)+dµ = ✶ ✷

• G∗∗

ˆ f d µ − t

• G∗

fdµ , ❢♦r t❤✐s ❝❤♦✐❝❡ ♦❢ f ✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✸✻ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r ●❛❧t♦♥✲❲❛ts♦♥ tr❡❡s

• ✐✈❡♥ ❛ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ {π(i) ,

i ≥ ✵} ♦♥ t❤❡ ♥♦♥♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs✱ ✇✐t❤ ✜♥✐t❡ ♠❡❛♥

i iπ(i)✱ ❞❡✜♥❡

ˆ π(i) := (i + ✶)π(i + ✶)

• i iπ(i)

, i ≥ ✵ . {ˆ π(i) , i ≥ ✵} ✐s ❛❧s♦ ❛ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✳ ❚❤❡ ✉♥✐♠♦❞✉❧❛r ●❛❧t♦♥✲❲❛ts♦♥ tr❡❡✱ ❯●❲❚✭π✮ ✐s t❤❡ r❛♥❞♦♠ tr❡❡ ❝♦♥str✉❝t❡❞ ❛s ❢♦❧❧♦✇s✿ ❙t❛rt ✇✐t❤ ❛ r♦♦t ❛♥❞ ❣✐✈❡ ✐t ❛ r❛♥❞♦♠ ♥✉♠❜❡r ♦❢ ❝❤✐❧❞r❡♥ ✭❛t ❞❡♣t❤ ✶✮ ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ❝❤✐❧❞r❡♥ ❞✐str✐❜✉t❡❞ ❛s π✳ ❋♦r ❡❛❝❤ ❝❤✐❧❞✱ ❣✐✈❡ ✐t ❛ r❛♥❞♦♠ ♥✉♠❜❡r ♦❢ ❝❤✐❧❞r❡♥ ✭❛t ❞❡♣t❤ ✷✮✱ t❤❡ ♥✉♠❜❡r ❞✐str✐❜✉t❡❞ ❛s ˆ π✱ ✐♥❞❡♣❡♥❞❡♥t❧②✳ ❘❡♣❡❛t ✭✉s✐♥❣ ˆ π ❢r♦♠ ♥♦✇ ♦♥✮✳ ▼❛♥② st❛♥❞❛r❞ s❡q✉❡♥❝❡s ♦❢ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ♠♦❞❡❧s✱ s✉❝❤ ❛s t❤❡ ♣❛✐r✐♥❣ ♠♦❞❡❧ ❜❛s❡❞ ♦♥ ❤❛❧❢ ❡❞❣❡s ❛♥❞ ✜①❡❞ ❞❡❣r❡❡ ❞✐str✐❜✉t✐♦♥s ✇❤✐❝❤ s❤♦✇s ✉♣ ✐♥ t❤❡ t❤❡♦r② ♦❢ ▲❉P❈ ❝♦❞❡s✱ ❤❛✈❡ ❛ ✉♥✐♠♦❞✉❧❛r

• ❛❧t♦♥✲❲❛ts♦♥ tr❡❡ ❛s t❤❡✐r ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✸✼ ✴ ✼✵

❘❡❝✉rs✐✈❡ ❞✐str✐❜✉t✐♦♥❛❧ ❡q✉❛t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t ♦♥ ✉♥✐♠♦❞✉❧❛r ●❛❧t♦♥✲❲❛ts♦♥ tr❡❡s

■❢ µ ✐s t❤❡ ❧❛✇ ♦❢ ❯●❲❚✭π✮✱ t❤❡♥ ❢♦r ❡✈❡r② t✱ ✇❡ ❤❛✈❡

Φµ(t) = ♠❛①

Q=Fπ,t(Q){E[D]

✷ P(ξ✶ + ξ✷ > ✶) − tP(ξ✶ + . . . + ξD > t)} ,

✇❤❡r❡ Fπ,t(Q) ✐s t❤❡ ❧❛✇ ♦❢ [✶ − t + ξ✶ + . . . + ξ ˆ

D]✶ ✵✳

❍❡r❡ [a]✶

✵ ❡q✉❛❧s ✵ ✐❢ a < ✵✱ ✶ ✐❢ a > ✶ ❛♥❞ a ♦t❤❡r✇✐s❡✳ ❆❧s♦✱ ˆ

D ❤❛s t❤❡ ❧❛✇ ˆ π✱ D ❤❛s t❤❡ ❧❛✇ π✱ ❛♥❞ t❤❡ ξi ❛r❡ ✐✳✐✳❞✳ ✇✐t❤ ❧❛✇ Q✳ ❘❡❝❛❧❧ t❤❛t t → Φµ(t) :=

• G∗

(∂Θ✵ − t)+dµ , ❝❤❛r❛❝t❡r✐③❡s t❤❡ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♥❞✉❝❡❞ ❧♦❛❞ ❛t t❤❡ r♦♦t✳ ❚❤❡ ❛❜♦✈❡ r❡❝✉rs✐✈❡ ❞✐str✐❜✉t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ✐s ✐♥ ❡✛❡❝t t❤❡ ♦♥❡ ❝♦♥❥❡❝t✉r❡❞ ❜② ❍❛❥❡❦✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✸✽ ✴ ✼✵

■♥t✉✐t✐♦♥ ❜❡❤✐♥❞ t❤❡ ❘❉❊

❲❡ ❝♦♥s✐❞❡r t❤❡ ❘❉❊ Q = Fπ,t(Q)✱ ✇❤❡r❡ Fπ,t(Q) ✐s t❤❡ ❧❛✇ ♦❢ [✶ − t + ξ✶ + . . . + ξ ˆ

D]✶ ✵✱ ✇❤❡r❡ ξ✶, ξ✷, . . . ❛r❡ ✐✳✐✳❞ ✇✐t❤ t❤❡ ❧❛✇ Q✳

❈♦♥s✐❞❡r ❛♥ ❡❞❣❡ (i, o)✳ ❲❡ ❛r❡ ✏s♦❧✈✐♥❣ ❢♦r t❤❡ ❧♦❛❞ t❤❛t ♣❛ss❡s ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ❢r♦♠ o t♦ i✳ ❋♦r ✶ ≤ k ≤ ˆ D✱ ✶ − ξk ❤❛s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❛♠♦✉♥t ♦❢ ❧♦❛❞ t❤❛t ❝❛♥ ❜❡ ❛❜s♦r❜❡❞ ❜② t❤❡ k✲t❤ ❝❤✐❧❞ ♦❢ o ✭t❤✐♥❦ ♦❢ i ❛s t❤❡ ♣❛r❡♥t ♦❢ o ❛♥❞ ♥♦t ❛s ❛ ❝❤✐❧❞✮✱ t❤✐s ❝❤✐❧❞ ♦❢ ❝♦✉rs❡ s✉♣♣♦rt✐♥❣ ✐ts ♦✇♥ s✉❜tr❡❡ ♦❢ ❝❤✐❧❞r❡♥✱ s✉❝❤ ❛s t♦ ♠❛❦❡ t❤❡ ♥❡t ❧♦❛❞ ❛t t❤❛t ❝❤✐❧❞ ❡q✉❛❧ t♦ t✳ ❚❤❡ ♥✉♠❜❡r [✶ − (t − ξ✶ − . . . − ξ ˆ

D)]✶ ✵ ✐s t❤❡♥ t❤❡ ❛♠♦✉♥t t❤❛t

✇♦✉❧❞ ❜❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ❢r♦♠ ♥♦❞❡ o t♦ ♥♦❞❡ i ✐♥ ♦r❞❡r t♦ ♠❛✐♥t❛✐♥ ❛ t♦t❛❧ ❧♦❛❞ ♦❢ t ❛t ♥♦❞❡ o✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✸✾ ✴ ✼✵

❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♠❛①✐♠✉♠ ❧♦❛❞

❯♥❞❡r ❛ ♠✐❧❞ ❛❞❞✐t✐♦♥❛❧ ♦♥ t❤❡ ❞❡❣r❡❡ ❞✐str✐❜✉t✐♦♥s t❤❡ ♠❛①✐♠✉♠ ❧♦❛❞ ❛❧s♦ ❝♦♥✈❡r❣❡s t♦ t❤❡ ♠❛①✐♠✉♠ ♦❢ t❤❡ ❧✐♠✐t✳ ❚❤✐s ✈❡r✐✜❡s t❤❡ ❝♦♥❥❡❝t✉r❡ ♦❢ ❍❛❥❡❦ r❡❣❛r❞✐♥❣ t❤❡ ❧✐♠✐t ♦❢ t❤❡ ♠❛①✐♠✉♠ ❧♦❛❞✳ ❖♥❡ ♠✉st ❡①❝❧✉❞❡ ✏❧♦❝❛❧ ♣♦❝❦❡ts ♦❢ ❤✐❣❤ ❡❞❣❡ ❞❡♥s✐t②✧ ✐♥ t❤❡ ❣r❛♣❤✳ ❆ss✉♠❡ t❤❛t ❢♦r s♦♠❡ λ > ✵ ✇❡ ❤❛✈❡ s✉♣

n≥✶

{✶ n

n

• i=✶

eλdn(i)} < ∞ . ▲❡t Z (n)

δ,t ❞❡♥♦t❡ t❤❡ ♥✉♠❜❡r ♦❢ s✉❜s❡ts S ♦❢ {✶, . . . , n} ♦❢ s✐③❡

|S| ≤ δn ✇✐t❤ ❡❞❣❡ ❝♦✉♥t |E(S)| ≥ t|S| ✐♥ t❤❡ ❣✐✈❡♥ r❛♥❞♦♠ ♣❛✐r✐♥❣ ♠♦❞❡❧✳ ❚❤❡♥ ✇❡ ❝❛♥ s❤♦✇ t❤❛t P(Z (n)

δ,t > ✵) → ✵ ,

❛s n → ∞ . ❚❤✐s s✉✣❝❡s✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✵ ✴ ✼✵

❙❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ♠❛✐♥ r❡s✉❧t

❚❤❡ ❦❡② ✐❞❡❛ ✐s t♦ ❝♦♥s✐❞❡r s♦✲❝❛❧❧❡❞ ǫ✲❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥s✱ ✐✳❡✳ ❛❧❧♦❝❛t✐♦♥s θ ♦♥ ❛ ❧♦❝❛❧❧② ✜♥✐t❡ ❣r❛♣❤ G t❤❛t s❛t✐s❢② θ(i, j) = ✶ ✷ + ✶ ✷ǫ(∂θ(i) − ∂θ(j)) ✶

✵

. ❚❤❡r❡ ✐s ❛ ❜✉✐❧t✲✐♥ ❝♦♥tr❛❝t✐✈✐t② ✐♥ t❤✐s ❞❡✜♥✐t✐♦♥ ❢♦r ❜♦✉♥❞❡❞ ❞❡❣r❡❡ ❣r❛♣❤s✱ ✇❤✐❝❤ ❛❧❧♦✇s ♦♥❡ t♦ ❡st❛❜❧✐s❤ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ ǫ✲❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥s ❢♦r s✉❝❤ ❣r❛♣❤s✳ ❚❤❡ ❝❛s❡ ♦❢ ❧♦❝❛❧❧② ✜♥✐t❡ ❣r❛♣❤s ❝❛♥ ❜❡ ❤❛♥❞❧❡❞ ❜② ❛ tr✉♥❝❛t✐♦♥ ❛r❣✉♠❡♥t✳ ❚❤❡ ❝❧❛✐♠❡❞ Θ✵ ❝❛♥ t❤❡♥ ❜❡ s❤♦✇♥ t♦ ❡①✐st ❛s ❛ ❧✐♠✐t ✐♥ L✷ ♦❢ t❤❡ ǫ✲❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥s ❛s ǫ → ✵✳ ❚❤❡ ǫ✲r❡❧❛①❛t✐♦♥ ❝❛♥ ❜❡ r♦✉❣❤❧② t❤♦✉❣❤t ♦❢ ❛s ❛♥❛❧♦❣♦✉s t♦ ✇♦r❦✐♥❣ ❛t ✜♥✐t❡ t❡♠♣❡r❛t✉r❡ ✭✈❡rs✉s ③❡r♦ t❡♠♣❡r❛t✉r❡✮ ✐♥ st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✶ ✴ ✼✵

❖✉t❧✐♥❡

✶

❆ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠ st✉❞✐❡❞ ❜② ❍❛❥❡❦

✷

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❣r❛♣❤s

✸

❚❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❧♦❝❛❧ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡

✹

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❤②♣❡r❣r❛♣❤s

✺

• r❛♣❤ ✐♥❞❡①❡❞ ❞❛t❛

✻

❯♥✐✈❡rs❛❧ ❝♦♠♣r❡ss✐♦♥ ♦❢ ❣r❛♣❤✐❝❛❧ ❞❛t❛

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✷ ✴ ✼✵

▲♦❛❞ ❇❛❧❛♥❝✐♥❣ ♦♥ ❛ ❤②♣❡r❣r❛♣❤

v✶ v✷ v✸ v✹ e✶ e✷ e✸ ✶ / ✷ ✵ ✶/✷ ✶ ✵ ✵ ✶ v✶ v✷ v✸ v✹ e✶ e✷ e✸ θ(e✶, v✶) = ✶/✷ ✵ ✶/✷ ✶ ✵ ✵ ✶

∂θ(✶) = ✶

✷

∂θ(✸) = ✶

✷

∂θ(✷) = ✶ ∂θ(✹) = ✶

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✸ ✴ ✼✵

H∗ ❛♥❞ H∗∗

i H∗ = {[H, i]} ❙✐♠♣❧❡✱ ❝♦♥♥❡❝t❡❞✱ ✜♥✐t❡ ❡❞❣❡s✱ ❧♦❝❛❧❧② ✜♥✐t❡

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✹ ✴ ✼✵

H∗ ❛♥❞ H∗∗

i H∗ = {[H, i]} e i H∗∗ = {[H, e, i]} ❙✐♠♣❧❡✱ ❝♦♥♥❡❝t❡❞✱ ✜♥✐t❡ ❡❞❣❡s✱ ❧♦❝❛❧❧② ✜♥✐t❡

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✹ ✴ ✼✵

H∗ ❛♥❞ H∗∗

i H∗ = {[H, i]} e i H∗∗ = {[H, e, i]} ❙✐♠♣❧❡✱ ❝♦♥♥❡❝t❡❞✱ ✜♥✐t❡ ❡❞❣❡s✱ ❧♦❝❛❧❧② ✜♥✐t❡

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✹ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r✐t②

❋✐♥✐t❡ Hn U(H) = ✶ |V (H)|

• i∈V (H)

δ[H,i] ∈ P(H∗) ✇❤❡♥ ◆♦t ❛❧❧ ❝❛♥ ❜❡ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐ts ♦❢ ✜♥✐t❡ ❤②♣❡r❣r❛♣❤s ❋♦r ✱ ❧❡t

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✺ ✴ ✼✵ ❯

❯♥✐♠♦❞✉❧❛r✐t②

❋✐♥✐t❡ Hn U(H) = ✶ |V (H)|

• i∈V (H)

δ[H,i] ∈ P(H∗) Hn

lwc

→ µ ✇❤❡♥ U(Hn) ⇒ µ ◆♦t ❛❧❧ ❝❛♥ ❜❡ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐ts ♦❢ ✜♥✐t❡ ❤②♣❡r❣r❛♣❤s ❋♦r ✱ ❧❡t

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✺ ✴ ✼✵ ❯

❯♥✐♠♦❞✉❧❛r✐t②

❋✐♥✐t❡ Hn U(H) = ✶ |V (H)|

• i∈V (H)

δ[H,i] ∈ P(H∗) Hn

lwc

→ µ ✇❤❡♥ U(Hn) ⇒ µ ◆♦t ❛❧❧ µ ❝❛♥ ❜❡ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐ts ♦❢ ✜♥✐t❡ ❤②♣❡r❣r❛♣❤s ❋♦r ✱ ❧❡t

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✺ ✴ ✼✵ ❯

❯♥✐♠♦❞✉❧❛r✐t②

❋✐♥✐t❡ Hn U(H) = ✶ |V (H)|

• i∈V (H)

δ[H,i] ∈ P(H∗) Hn

lwc

→ µ ✇❤❡♥ U(Hn) ⇒ µ ◆♦t ❛❧❧ µ ❝❛♥ ❜❡ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐ts ♦❢ ✜♥✐t❡ ❤②♣❡r❣r❛♣❤s ❋♦r f : H∗∗ → R✱ ❧❡t ∂f : H∗ → R ∂f (H, i) =

• e∋i

f (H, e, i)

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✺ ✴ ✼✵ ❯

❯♥✐♠♦❞✉❧❛r✐t② ✭❝♦♥t✬❞✮

❋♦r µ ∈ P(H∗)✱ ❞❡✜♥❡ µ ∈ M(H∗∗) ❛s

• fd

µ =

• ∂fdµ

❢♦r ❛❧❧ ❇♦r❡❧ ❢✉♥❝t✐♦♥ f ♦♥ H∗∗✳ ❋♦r ✱ ❧❡t ✶ ✐s ❝❛❧❧❡❞ ✉♥✐♠♦❞✉❧❛r ✐❢ ■❢ ✱ ✐s ✉♥✐♠♦❞✉❧❛r

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✻ ✴ ✼✵ ❯

❯♥✐♠♦❞✉❧❛r✐t② ✭❝♦♥t✬❞✮

❋♦r µ ∈ P(H∗)✱ ❞❡✜♥❡ µ ∈ M(H∗∗) ❛s

• fd

µ =

• ∂fdµ

❢♦r ❛❧❧ ❇♦r❡❧ ❢✉♥❝t✐♦♥ f ♦♥ H∗∗✳ ❋♦r f : H∗∗ → R✱ ❧❡t ∇f : H∗∗ → R ∇f (H, e, i) = ✶ |e|

• j∈e

f (H, e, j). ✐s ❝❛❧❧❡❞ ✉♥✐♠♦❞✉❧❛r ✐❢ ■❢ ✱ ✐s ✉♥✐♠♦❞✉❧❛r

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✻ ✴ ✼✵ ❯

❯♥✐♠♦❞✉❧❛r✐t② ✭❝♦♥t✬❞✮

❋♦r µ ∈ P(H∗)✱ ❞❡✜♥❡ µ ∈ M(H∗∗) ❛s

• fd

µ =

• ∂fdµ

❢♦r ❛❧❧ ❇♦r❡❧ ❢✉♥❝t✐♦♥ f ♦♥ H∗∗✳ ❋♦r f : H∗∗ → R✱ ❧❡t ∇f : H∗∗ → R ∇f (H, e, i) = ✶ |e|

• j∈e

f (H, e, j). µ ∈ P(H∗) ✐s ❝❛❧❧❡❞ ✉♥✐♠♦❞✉❧❛r ✐❢

• fd

µ =

• ∇fd

µ ■❢ ✱ ✐s ✉♥✐♠♦❞✉❧❛r

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✻ ✴ ✼✵ ❯

❯♥✐♠♦❞✉❧❛r✐t② ✭❝♦♥t✬❞✮

❋♦r µ ∈ P(H∗)✱ ❞❡✜♥❡ µ ∈ M(H∗∗) ❛s

• fd

µ =

• ∂fdµ

❢♦r ❛❧❧ ❇♦r❡❧ ❢✉♥❝t✐♦♥ f ♦♥ H∗∗✳ ❋♦r f : H∗∗ → R✱ ❧❡t ∇f : H∗∗ → R ∇f (H, e, i) = ✶ |e|

• j∈e

f (H, e, j). µ ∈ P(H∗) ✐s ❝❛❧❧❡❞ ✉♥✐♠♦❞✉❧❛r ✐❢

• fd

µ =

• ∇fd

µ ■❢ Hn

lwc

→ µ✱ µ ✐s ✉♥✐♠♦❞✉❧❛r

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✻ ✴ ✼✵ ❯

❇♦r❡❧ ❆❧❧♦❝❛t✐♦♥s ❛♥❞ ❇❛❧❛♥❝❡❞♥❡ss

Θ : H∗∗ → [✵, ✶] ✐s ❝❛❧❧❡❞ ❛ ❇♦r❡❧ ❛❧❧♦❝❛t✐♦♥ ✐❢

• j∈e

Θ(H, e, j) = ✶ ∀[H, e, i] ∈ H∗∗ ✐s ❜❛❧❛♥❝❡❞ ✇✳r✳t✳ ✐❢ ❢♦r ✕❛❧♠♦st ❛❧❧ ✵

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✼ ✴ ✼✵

❇♦r❡❧ ❆❧❧♦❝❛t✐♦♥s ❛♥❞ ❇❛❧❛♥❝❡❞♥❡ss

Θ : H∗∗ → [✵, ✶] ✐s ❝❛❧❧❡❞ ❛ ❇♦r❡❧ ❛❧❧♦❝❛t✐♦♥ ✐❢

• j∈e

Θ(H, e, j) = ✶ ∀[H, e, i] ∈ H∗∗ Θ ✐s ❜❛❧❛♥❝❡❞ ✇✳r✳t✳ µ ∈ P(H∗) ✐❢ ❢♦r µ✕❛❧♠♦st ❛❧❧ [H, e, i] ∈ H∗∗ j ∈ e ∂Θ(H, i) > ∂Θ(H, j) ⇒ Θ(H, e, i) = ✵.

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✼ ✴ ✼✵

▼❛✐♥ r❡s✉❧ts ✭✇✐t❤ P❛②❛♠ ❉❡❧❣♦s❤❛✮

❚❤❡♦r❡♠

❚❛❦❡ µ ∈ P(H∗) ✉♥✐♠♦❞✉❧❛r✱ ❞❡❣(µ), ❱❛r(µ) < ∞✱ t❤❡♥

✶

✭❡①✐st❡♥❝❡✮ ❛ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥

✵

✷

✭✉♥✐q✉❡♥❡ss✮

✶ ✷ t✇♦ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥s✱ t❤❡♥ ✶ ✷✱

✕❛✳s✳

✸

✭❝♦♥t✐♥✉✐t②✮ t❤❡♥

✹

✭♦♣t✐♠❛❧✐t②✮ ✐s ❜❛❧❛♥❝❡❞ ✐✛ ✐t ♠✐♥✐♠✐③❡s ❢♦r str✐❝t❧② ❝♦♥✈❡① ✵ ✳

✺

✭✈❛r✐❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❛♥❞ ❜❛❧❛♥❝❡❞✱ t❤❡♥ ♠❛①

❇♦r❡❧ ✵ ✶

♠✐♥

✇❤❡r❡

♠✐♥ ✶ ♠✐♥

✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✽ ✴ ✼✵ ❍❛❥❡❦ ❈❊ ❱❈

▼❛✐♥ r❡s✉❧ts ✭✇✐t❤ P❛②❛♠ ❉❡❧❣♦s❤❛✮

❚❤❡♦r❡♠

❚❛❦❡ µ ∈ P(H∗) ✉♥✐♠♦❞✉❧❛r✱ ❞❡❣(µ), ❱❛r(µ) < ∞✱ t❤❡♥

✶

✭❡①✐st❡♥❝❡✮ ∃ ❛ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ Θ✵

✷

✭✉♥✐q✉❡♥❡ss✮

✶ ✷ t✇♦ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥s✱ t❤❡♥ ✶ ✷✱

✕❛✳s✳

✸

✭❝♦♥t✐♥✉✐t②✮ t❤❡♥

✹

✭♦♣t✐♠❛❧✐t②✮ ✐s ❜❛❧❛♥❝❡❞ ✐✛ ✐t ♠✐♥✐♠✐③❡s ❢♦r str✐❝t❧② ❝♦♥✈❡① ✵ ✳

✺

✭✈❛r✐❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❛♥❞ ❜❛❧❛♥❝❡❞✱ t❤❡♥ ♠❛①

❇♦r❡❧ ✵ ✶

♠✐♥

✇❤❡r❡

♠✐♥ ✶ ♠✐♥

✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✽ ✴ ✼✵ ❍❛❥❡❦ ❈❊ ❱❈

▼❛✐♥ r❡s✉❧ts ✭✇✐t❤ P❛②❛♠ ❉❡❧❣♦s❤❛✮

❚❤❡♦r❡♠

❚❛❦❡ µ ∈ P(H∗) ✉♥✐♠♦❞✉❧❛r✱ ❞❡❣(µ), ❱❛r(µ) < ∞✱ t❤❡♥

✶

✭❡①✐st❡♥❝❡✮ ∃ ❛ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ Θ✵

✷

✭✉♥✐q✉❡♥❡ss✮ Θ✶, Θ✷ t✇♦ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥s✱ t❤❡♥ ∂Θ✶ = ∂Θ✷✱ µ✕❛✳s✳

✸

✭❝♦♥t✐♥✉✐t②✮ t❤❡♥

✹

✭♦♣t✐♠❛❧✐t②✮ ✐s ❜❛❧❛♥❝❡❞ ✐✛ ✐t ♠✐♥✐♠✐③❡s ❢♦r str✐❝t❧② ❝♦♥✈❡① ✵ ✳

✺

✭✈❛r✐❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❛♥❞ ❜❛❧❛♥❝❡❞✱ t❤❡♥ ♠❛①

❇♦r❡❧ ✵ ✶

♠✐♥

✇❤❡r❡

♠✐♥ ✶ ♠✐♥

✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✽ ✴ ✼✵ ❍❛❥❡❦ ❈❊ ❱❈

▼❛✐♥ r❡s✉❧ts ✭✇✐t❤ P❛②❛♠ ❉❡❧❣♦s❤❛✮

❚❤❡♦r❡♠

❚❛❦❡ µ ∈ P(H∗) ✉♥✐♠♦❞✉❧❛r✱ ❞❡❣(µ), ❱❛r(µ) < ∞✱ t❤❡♥

✶

✭❡①✐st❡♥❝❡✮ ∃ ❛ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ Θ✵

✷

✭✉♥✐q✉❡♥❡ss✮ Θ✶, Θ✷ t✇♦ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥s✱ t❤❡♥ ∂Θ✶ = ∂Θ✷✱ µ✕❛✳s✳

✸

✭❝♦♥t✐♥✉✐t②✮ Hn

lwc

→ µ t❤❡♥ Ln ⇒ L

✹

✭♦♣t✐♠❛❧✐t②✮ ✐s ❜❛❧❛♥❝❡❞ ✐✛ ✐t ♠✐♥✐♠✐③❡s ❢♦r str✐❝t❧② ❝♦♥✈❡① ✵ ✳

✺

✭✈❛r✐❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❛♥❞ ❜❛❧❛♥❝❡❞✱ t❤❡♥ ♠❛①

❇♦r❡❧ ✵ ✶

♠✐♥

✇❤❡r❡

♠✐♥ ✶ ♠✐♥

✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✽ ✴ ✼✵ ❍❛❥❡❦ ❈❊ ❱❈

▼❛✐♥ r❡s✉❧ts ✭✇✐t❤ P❛②❛♠ ❉❡❧❣♦s❤❛✮

❚❤❡♦r❡♠

❚❛❦❡ µ ∈ P(H∗) ✉♥✐♠♦❞✉❧❛r✱ ❞❡❣(µ), ❱❛r(µ) < ∞✱ t❤❡♥

✶

✭❡①✐st❡♥❝❡✮ ∃ ❛ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ Θ✵

✷

✭✉♥✐q✉❡♥❡ss✮ Θ✶, Θ✷ t✇♦ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥s✱ t❤❡♥ ∂Θ✶ = ∂Θ✷✱ µ✕❛✳s✳

✸

✭❝♦♥t✐♥✉✐t②✮ Hn

lwc

→ µ t❤❡♥ Ln ⇒ L

✹

✭♦♣t✐♠❛❧✐t②✮ Θ ✐s ❜❛❧❛♥❝❡❞ ✐✛ ✐t ♠✐♥✐♠✐③❡s

• f (∂Θ)dµ ❢♦r str✐❝t❧② ❝♦♥✈❡①

f : [✵, ∞) → R✳

✺

✭✈❛r✐❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❛♥❞ ❜❛❧❛♥❝❡❞✱ t❤❡♥ ♠❛①

❇♦r❡❧ ✵ ✶

♠✐♥

✇❤❡r❡

♠✐♥ ✶ ♠✐♥

✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✽ ✴ ✼✵ ❍❛❥❡❦ ❈❊ ❱❈

▼❛✐♥ r❡s✉❧ts ✭✇✐t❤ P❛②❛♠ ❉❡❧❣♦s❤❛✮

❚❤❡♦r❡♠

❚❛❦❡ µ ∈ P(H∗) ✉♥✐♠♦❞✉❧❛r✱ ❞❡❣(µ), ❱❛r(µ) < ∞✱ t❤❡♥

✶

✭❡①✐st❡♥❝❡✮ ∃ ❛ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ Θ✵

✷

✭✉♥✐q✉❡♥❡ss✮ Θ✶, Θ✷ t✇♦ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥s✱ t❤❡♥ ∂Θ✶ = ∂Θ✷✱ µ✕❛✳s✳

✸

✭❝♦♥t✐♥✉✐t②✮ Hn

lwc

→ µ t❤❡♥ Ln ⇒ L

✹

✭♦♣t✐♠❛❧✐t②✮ Θ ✐s ❜❛❧❛♥❝❡❞ ✐✛ ✐t ♠✐♥✐♠✐③❡s

• f (∂Θ)dµ ❢♦r str✐❝t❧② ❝♦♥✈❡①

f : [✵, ∞) → R✳

✺

✭✈❛r✐❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✮ t ∈ R ❛♥❞ Θ ❜❛❧❛♥❝❡❞✱ t❤❡♥

• (∂Θ − t)+dµ =

♠❛①

f ∈H∗

❇♦r❡❧

→ [✵,✶]

• ˜

f♠✐♥d µ − t

• fdµ

✇❤❡r❡ ˜ f♠✐♥(H, e, i) =

✶ |e| ♠✐♥j∈e f (H, j)✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✽ ✴ ✼✵ ❍❛❥❡❦ ❈❊ ❱❈

❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i ✳ ✳ ✳ ✳ ✳ ✳ ρT,i(x) = t♦t❛❧ ❧♦❛❞ ❛t i ✇✐t❤ ❜❛s❡❧♦❛❞ x ✳ ✳ ✳ ✳ ✳ ✳

✶

✿ t❤❡ ❛♠♦✉♥t ♦❢ ❡①tr❛ ❧♦❛❞ s♦ t❤❛t t❤❡ t♦t❛❧ ❧♦❛❞ ❜❡❝♦♠❡s

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✾ ✴ ✼✵

❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i ✳ ✳ ✳ ✳ ✳ ✳ ↓ x ρT,i(x) = t♦t❛❧ ❧♦❛❞ ❛t i ✇✐t❤ ❜❛s❡❧♦❛❞ x ✳ ✳ ✳ ✳ ✳ ✳

✶

✿ t❤❡ ❛♠♦✉♥t ♦❢ ❡①tr❛ ❧♦❛❞ s♦ t❤❛t t❤❡ t♦t❛❧ ❧♦❛❞ ❜❡❝♦♠❡s

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✾ ✴ ✼✵

❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i ✳ ✳ ✳ ✳ ✳ ✳ ↓ x ρT,i(x) = t♦t❛❧ ❧♦❛❞ ❛t i ✇✐t❤ ❜❛s❡❧♦❛❞ x ✳ ✳ ✳ ✳ ✳ ✳

✶

✿ t❤❡ ❛♠♦✉♥t ♦❢ ❡①tr❛ ❧♦❛❞ s♦ t❤❛t t❤❡ t♦t❛❧ ❧♦❛❞ ❜❡❝♦♠❡s

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✾ ✴ ✼✵

❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i ✳ ✳ ✳ ✳ ✳ ✳ ↓ x ρT,i(x) = t♦t❛❧ ❧♦❛❞ ❛t i ✇✐t❤ ❜❛s❡❧♦❛❞ x x ρ(x) ✳ ✳ ✳ ✳ ✳ ✳

✶

✿ t❤❡ ❛♠♦✉♥t ♦❢ ❡①tr❛ ❧♦❛❞ s♦ t❤❛t t❤❡ t♦t❛❧ ❧♦❛❞ ❜❡❝♦♠❡s

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✾ ✴ ✼✵

❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i ✳ ✳ ✳ ✳ ✳ ✳ ↓ x ρT,i(x) = t♦t❛❧ ❧♦❛❞ ❛t i ✇✐t❤ ❜❛s❡❧♦❛❞ x x ρ(x) i ✳ ✳ ✳ ✳ ✳ ✳ ↓ ? t ρ−✶

T,i(t)✿ t❤❡ ❛♠♦✉♥t ♦❢ ❡①tr❛ ❧♦❛❞

s♦ t❤❛t t❤❡ t♦t❛❧ ❧♦❛❞ ❜❡❝♦♠❡s t

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✾ ✴ ✼✵

❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i

j✷ j✶ j✸ j✹

e✶ e✷ Te✶,j✷ Te✶,j✶ Te✷,j✸ Te✷,j✹

✶ ✶ ✶ ✶

✶ ✶

✶

✶

✶

✶

✶

✶

✶

✶ ✶

✶

✶ ✶ ✵

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i

j✷ j✶ j✸ j✹

e✶ e✷ Te✶,j✷ Te✶,j✶ Te✷,j✸ Te✷,j✹

ρ−✶

T,i(t) =? ✶ ✶ ✶

✶ ✶

✶

✶

✶

✶

✶

✶

✶

✶ ✶

✶

✶ ✶ ✵

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i

j✷ j✶ j✸ j✹

e✶ e✷ Te✶,j✷ Te✶,j✶ Te✷,j✸ Te✷,j✹

ρ−✶

T,i(t) =?

↓ ? t

✶ ✶ ✶

✶ ✶

✶

✶

✶

✶

✶

✶

✶

✶ ✶

✶

✶ ✶ ✵

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i

j✷ j✶ j✸ j✹

e✶ e✷ Te✶,j✷ Te✶,j✶ Te✷,j✸ Te✷,j✹

ρ−✶

T,i(t) =?

↓ ? t

t t t t

✶ ✶ ✶

✶ ✶

✶

✶

✶

✶

✶

✶

✶

✶ ✶

✶

✶ ✶ ✵

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i

j✷ j✶ j✸ j✹

e✶ e✷ Te✶,j✷ Te✶,j✶ Te✷,j✸ Te✷,j✹

ρ−✶

T,i(t) =?

↓ ? t

t t t t

θ(e✶, j✶) = ρ−✶

Te✶,j✶(t) ✶

✶

✶

✶

✶

✶

✶

✶ ✶

✶

✶ ✶ ✵

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i

j✷ j✶ j✸ j✹

e✶ e✷ Te✶,j✷ Te✶,j✶ Te✷,j✸ Te✷,j✹

ρ−✶

T,i(t) =?

↓ ? t

t t t t

θ(e✶, j✶) = ρ−✶

Te✶,j✶(t) ✶

✶

✶

✶

✶

✶

✶

✶ ✶

✶

✶ ✶ ✵

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i

j✷ j✶ j✸ j✹

e✶ e✷ Te✶,j✷ Te✶,j✶ Te✷,j✸ Te✷,j✹

ρ−✶

T,i(t) =?

↓ ? t

t t t t

θ(e✶, j✶) = ρ−✶

Te✶,j✶(t)

θ(e✶, i) = ✶ −

j∈e✶

j=i ρ−✶ Te✶,j(t)

✶

✶

✶ ✶

✶

✶ ✶ ✵

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i

j✷ j✶ j✸ j✹

e✶ e✷ Te✶,j✷ Te✶,j✶ Te✷,j✸ Te✷,j✹

ρ−✶

T,i(t) =?

↓ ? t

t t t t

θ(e✶, j✶) = ρ−✶

Te✶,j✶(t)

θ(e✶, i) = ✶ −

j∈e✶

j=i ρ−✶ Te✶,j(t)

ρ−✶

T,i(t) = t −

• e∋i
• ✶ −
• j∈e

j=i

ρ−✶

Te,j(t)

• ✶

✶

✶ ✶ ✵

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥

i

j✷ j✶ j✸ j✹

e✶ e✷ Te✶,j✷ Te✶,j✶ Te✷,j✸ Te✷,j✹

ρ−✶

T,i(t) =?

↓ ? t

t t t t

θ(e✶, j✶) = ρ−✶

Te✶,j✶(t)

θ(e✶, i) = ✶ −

j∈e✶

j=i ρ−✶ Te✶,j(t)

ρ−✶

T,i(t) = t −

• e∋i
• ✶ −
• j∈e

j=i

ρ−✶

Te,j(t)

• ρ−✶

T,i(t) = t −

• e∋i
• ✶ −
• j∈e

j=i

ρ−✶

Te,j(t)+

✶

✵

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r ●❛❧t♦♥ ❲❛ts♦♥ ❍②♣❡rtr❡❡s

❆❧❧ t❤❡ ❤②♣❡r❡❞❣❡s ❤❛✈❡ s✐③❡ c ✭s❛② ✸✮ ❞✐str✐❜✉t✐♦♥ ♦♥ ♥♦♥✕♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶

✶

❯●❲❚ ✐s ✉♥✐♠♦❞✉❧❛r

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✶ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r ●❛❧t♦♥ ❲❛ts♦♥ ❍②♣❡rtr❡❡s

❆❧❧ t❤❡ ❤②♣❡r❡❞❣❡s ❤❛✈❡ s✐③❡ c ✭s❛② ✸✮ ❞✐str✐❜✉t✐♦♥ P ♦♥ ♥♦♥✕♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶

✶

❯●❲❚ ✐s ✉♥✐♠♦❞✉❧❛r

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✶ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r ●❛❧t♦♥ ❲❛ts♦♥ ❍②♣❡rtr❡❡s

❆❧❧ t❤❡ ❤②♣❡r❡❞❣❡s ❤❛✈❡ s✐③❡ c ✭s❛② ✸✮ ❞✐str✐❜✉t✐♦♥ P ♦♥ ♥♦♥✕♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶

✶

❯●❲❚ ✐s ✉♥✐♠♦❞✉❧❛r

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✶ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r ●❛❧t♦♥ ❲❛ts♦♥ ❍②♣❡rtr❡❡s

❆❧❧ t❤❡ ❤②♣❡r❡❞❣❡s ❤❛✈❡ s✐③❡ c ✭s❛② ✸✮ ❞✐str✐❜✉t✐♦♥ P ♦♥ ♥♦♥✕♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶

✶

❯●❲❚ ✐s ✉♥✐♠♦❞✉❧❛r

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✶ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r ●❛❧t♦♥ ❲❛ts♦♥ ❍②♣❡rtr❡❡s

❆❧❧ t❤❡ ❤②♣❡r❡❞❣❡s ❤❛✈❡ s✐③❡ c ✭s❛② ✸✮ ❞✐str✐❜✉t✐♦♥ P ♦♥ ♥♦♥✕♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs P

ˆ P

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ˆ Pk = (k+✶)Pk+✶

E[P]

❯●❲❚ ✐s ✉♥✐♠♦❞✉❧❛r

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✶ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r ●❛❧t♦♥ ❲❛ts♦♥ ❍②♣❡rtr❡❡s

❆❧❧ t❤❡ ❤②♣❡r❡❞❣❡s ❤❛✈❡ s✐③❡ c ✭s❛② ✸✮ ❞✐str✐❜✉t✐♦♥ P ♦♥ ♥♦♥✕♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs P

ˆ P

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ˆ Pk = (k+✶)Pk+✶

E[P]

❯●❲❚ ✐s ✉♥✐♠♦❞✉❧❛r

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✶ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r ●❛❧t♦♥ ❲❛ts♦♥ ❍②♣❡rtr❡❡s

❆❧❧ t❤❡ ❤②♣❡r❡❞❣❡s ❤❛✈❡ s✐③❡ c ✭s❛② ✸✮ ❞✐str✐❜✉t✐♦♥ P ♦♥ ♥♦♥✕♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs P

ˆ P

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ˆ Pk = (k+✶)Pk+✶

E[P]

❯●❲❚c(P) ∈ P(H∗) ✐s ✉♥✐♠♦❞✉❧❛r

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✶ ✴ ✼✵

▼❡❛♥ ❊①❝❡ss ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❯●❲❚c(P)

• (∂Θ − t)+dµ =

s✉♣

f :H∗→[✵,✶]

• ˜

f♠✐♥d µ − t

• fdµ

♦♣t✐♠❛❧ ✶ ✵

✶

✵

✶

❯●❲❚

✶

✶ ✶ ✶ ✶ ✶ ✵

✵

✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✷ ✴ ✼✵

▼❡❛♥ ❊①❝❡ss ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❯●❲❚c(P)

• (∂Θ − t)+dµ =

s✉♣

f :H∗→[✵,✶]

• ˜

f♠✐♥d µ − t

• fdµ

♦♣t✐♠❛❧ f = ✶ [∂Θ > t] ✵

✶

✵

✶

❯●❲❚

✶

✶ ✶ ✶ ✶ ✶ ✵

✵

✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✷ ✴ ✼✵

▼❡❛♥ ❊①❝❡ss ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❯●❲❚c(P)

• (∂Θ − t)+dµ =

s✉♣

f :H∗→[✵,✶]

• ˜

f♠✐♥d µ − t

• fdµ

♦♣t✐♠❛❧ f = ✶ [∂Θ > t] ∂Θ > t ⇔ ρ(✵) > t ⇔ ρ−✶(t) < ✵

x ρ(x) ∂θ

ρ−✶(t) t ❯●❲❚

✶

✶ ✶ ✶ ✶ ✶ ✵

✵

✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✷ ✴ ✼✵

▼❡❛♥ ❊①❝❡ss ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❯●❲❚c(P)

• (∂Θ − t)+dµ =

s✉♣

f :H∗→[✵,✶]

• ˜

f♠✐♥d µ − t

• fdµ

♦♣t✐♠❛❧ f = ✶ [∂Θ > t] ∂Θ > t ⇔ ρ(✵) > t ⇔ ρ−✶(t) < ✵

x ρ(x) ∂θ

ρ−✶(t) t ❯●❲❚c(P) . . . . . .

✶ N

• fdµ = P
• t − N

i=✶

• ✶ − X +

i,✶ − · · · − X + i,c−✶

✶

✵ < ✵

• X✶,✶

X✶,c−✶ XN,c−✶

✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✷ ✴ ✼✵

▼❡❛♥ ❊①❝❡ss ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❯●❲❚c(P)

• (∂Θ − t)+dµ =

s✉♣

f :H∗→[✵,✶]

• ˜

f♠✐♥d µ − t

• fdµ

♦♣t✐♠❛❧ f = ✶ [∂Θ > t] ∂Θ > t ⇔ ρ(✵) > t ⇔ ρ−✶(t) < ✵

x ρ(x) ∂θ

ρ−✶(t) t ❯●❲❚c(P) . . . . . .

✶ N

• fdµ = P
• t − N

i=✶

• ✶ − X +

i,✶ − · · · − X + i,c−✶

✶

✵ < ✵

• X✶,✶

X✶,c−✶ XN,c−✶ . . . . . . X ′

✶,✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✷ ✴ ✼✵

▼❡❛♥ ❊①❝❡ss ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❯●❲❚c(P)

• (∂Θ − t)+dµ =

s✉♣

f :H∗→[✵,✶]

• ˜

f♠✐♥d µ − t

• fdµ

♦♣t✐♠❛❧ f = ✶ [∂Θ > t] ∂Θ > t ⇔ ρ(✵) > t ⇔ ρ−✶(t) < ✵

x ρ(x) ∂θ

ρ−✶(t) t ❯●❲❚c(P) . . . . . .

✶ N

• fdµ = P
• t − N

i=✶

• ✶ − X +

i,✶ − · · · − X + i,c−✶

✶

✵ < ✵

• X✶,✶

X✶,c−✶ XN,c−✶ . . . . . . X ′

✶,✶

X✶,✶ = t − ˆ

N j=✶

• ✶ − X

′+

j,✶ − · · · − X

′+

j,c−✶

✶

✵ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✷ ✴ ✼✵

▼❡❛♥ ❊①❝❡ss ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❯●❲❚c(P)

• (∂Θ − t)+dµ =

s✉♣

f :H∗→[✵,✶]

• ˜

f♠✐♥d µ − t

• fdµ

♦♣t✐♠❛❧ f = ✶ [∂Θ > t] ∂Θ > t ⇔ ρ(✵) > t ⇔ ρ−✶(t) < ✵

x ρ(x) ∂θ

ρ−✶(t) t ❯●❲❚c(P) . . . . . .

✶ N

• fdµ = P
• t − N

i=✶

• ✶ − X +

i,✶ − · · · − X + i,c−✶

✶

✵ < ✵

• X✶,✶

X✶,c−✶ XN,c−✶ . . . . . . X ′

✶,✶

X✶,✶ = t − ˆ

N j=✶

• ✶ − X

′+

j,✶ − · · · − X

′+

j,c−✶

✶

✵

Q = F c

P,t(Q) ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✷ ✴ ✼✵

▼❡❛♥ ❊①❝❡ss ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❯●❲❚c(P) ✭❝♦♥t✬❞✮

❚❤❡♦r❡♠

❆ss✉♠❡ P ✐s ❛ ❞✐str✐❜✉t✐♦♥ ♦♥ ♥♦♥♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ✇✐t❤ ✜♥✐t❡ ✈❛r✐❛♥❝❡ ❛♥❞ µ = ❯●❲❚c(P)✳ ❚❤❡♥✱ ✇❡ ❤❛✈❡

• (∂Θ − t)+dµ =

♠❛①

Q:F c

P,t(Q)=Q

E [N] c P

• X +

✶ + · · · + X + c < ✶

• − tP (Y✶ + · · · + YN > t) ,

✇❤❡r❡ N ❤❛s ❞✐str✐❜✉t✐♦♥ P✱ Xi✬s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ❤❛✈❡ ❞✐str✐❜✉t✐♦♥ Q ❛♥❞ Yi✬s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ❡❛❝❤ ❤❛✈❡ ❞✐str✐❜✉t✐♦♥ ♦❢ [✶ − (X +

✶ + · · · + X + c−✶)]✶ ✵ ✇❤❡r❡ Xi✬s ❛r❡ ✐✳✐✳❞✳ ❢r♦♠ Q✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✸ ✴ ✼✵

❖✉t❧✐♥❡

✶

❆ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠ st✉❞✐❡❞ ❜② ❍❛❥❡❦

✷

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❣r❛♣❤s

✸

❚❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❧♦❝❛❧ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡

✹

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❤②♣❡r❣r❛♣❤s

✺

• r❛♣❤ ✐♥❞❡①❡❞ ❞❛t❛

✻

❯♥✐✈❡rs❛❧ ❝♦♠♣r❡ss✐♦♥ ♦❢ ❣r❛♣❤✐❝❛❧ ❞❛t❛

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✹ ✴ ✼✵

❙♦✉r❝❡s ♦❢ ❜✐❣ ❣r❛♣❤✐❝❛❧ ❞❛t❛✿ ❚❤❡ ✇❡❜

≈ ✹✼ ❜✐❧❧✐♦♥ ✇❡❜♣❛❣❡s

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✺ ✴ ✼✵

❙♦✉r❝❡s ♦❢ ❜✐❣ ❣r❛♣❤✐❝❛❧ ❞❛t❛✿ ❙♦❝✐❛❧ ♥❡t✇♦r❦s

≈ ✶✳✽ ❜✐❧❧✐♦♥ ❛❝t✐✈❡ ✉s❡rs ♦♥ ❋❛❝❡❜♦♦❦

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✻ ✴ ✼✵

❙♦✉r❝❡s ♦❢ ❜✐❣ ❣r❛♣❤✐❝❛❧ ❞❛t❛✿ ❇✐♦❧♦❣✐❝❛❧ ♥❡t✇♦r❦s

✵✳✷✺ ♠✐❧❧✐♦♥ ✲ ✶ ♠✐❧❧✐♦♥ ❡st✐♠❛t❡❞ ❤✉♠❛♥ ♣r♦t❡✐♥s

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✼ ✴ ✼✵

❯♥✐✈❡rs❛❧ ❝♦♠♣r❡ss✐♦♥ ♦❢ ♠❛r❦❡❞ ❣r❛♣❤✐❝❛❧ ❞❛t❛

❲❡ ✇❛♥t t♦ ❝♦♠♣r❡ss ❞♦✇♥ t♦ t❤❡ ✏❡♥tr♦♣②✧ ♦❢ t❤❡ ❞❛t❛✳ ❯♥✐✈❡rs❛❧✐t② ♠❡❛♥s t❤❛t t❤❡ s❝❤❡♠❡ s❤♦✉❧❞ ✇♦r❦ ✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ✏st❛t✐st✐❝s✧ ♦❢ t❤❡ ❞❛t❛✳ ■❞❡❛❧❧②✱ t❤❡ ❝♦♠♣r❡ss❡❞ r❡♣r❡s❡♥t❛t✐♦♥ s❤♦✉❧❞ ❡♥❛❜❧❡ ❛♥❛❧②s✐s ❛♥❞ q✉❡r②✐♥❣ ✐♥ t❤❡ ❝♦♠♣r❡ss❡❞ ❢♦r♠ ❚❤❡ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t t❤❡♦r② ❛❧❧♦✇s ♦♥❡ t♦ ♣r❡❝✐s❡❧② ❢♦r♠✉❧❛t❡ t❤❡ ✉♥✐✈❡rs❛❧ ❝♦♠♣r❡ss✐♦♥ ♣r♦❜❧❡♠ ❛♥❞ t♦ ♣r♦✈✐❞❡ ❛ s♦❧✉t✐♦♥✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✽ ✴ ✼✵

❯♥✐✈❡rs❛❧ ❝♦♠♣r❡ss✐♦♥ ♦❢ ♠❛r❦❡❞ ❣r❛♣❤✐❝❛❧ ❞❛t❛

❲❡ ✇❛♥t t♦ ❝♦♠♣r❡ss ❞♦✇♥ t♦ t❤❡ ✏❡♥tr♦♣②✧ ♦❢ t❤❡ ❞❛t❛✳ ❯♥✐✈❡rs❛❧✐t② ♠❡❛♥s t❤❛t t❤❡ s❝❤❡♠❡ s❤♦✉❧❞ ✇♦r❦ ✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ✏st❛t✐st✐❝s✧ ♦❢ t❤❡ ❞❛t❛✳ ■❞❡❛❧❧②✱ t❤❡ ❝♦♠♣r❡ss❡❞ r❡♣r❡s❡♥t❛t✐♦♥ s❤♦✉❧❞ ❡♥❛❜❧❡ ❛♥❛❧②s✐s ❛♥❞ q✉❡r②✐♥❣ ✐♥ t❤❡ ❝♦♠♣r❡ss❡❞ ❢♦r♠ ❚❤❡ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t t❤❡♦r② ❛❧❧♦✇s ♦♥❡ t♦ ♣r❡❝✐s❡❧② ❢♦r♠✉❧❛t❡ t❤❡ ✉♥✐✈❡rs❛❧ ❝♦♠♣r❡ss✐♦♥ ♣r♦❜❧❡♠ ❛♥❞ t♦ ♣r♦✈✐❞❡ ❛ s♦❧✉t✐♦♥✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✽ ✴ ✼✵

❚❤❡ ❇❈ ❡♥tr♦♣②✿ ❝♦✉♥t✐♥❣ t②♣✐❝❛❧ ❣r❛♣❤s

Ξ✿ ❡❞❣❡ ♠❛r❦s✱ Θ✿ ✈❡rt❡① ♠❛r❦s✱ ❜♦t❤ ✜♥✐t❡ G(n)

♠n,✉n✿ s❡t ♦❢ ❣r❛♣❤s ♦♥ n ✈❡rt✐❝❡s ✇✐t❤ ♠n(x) ♠❛♥② ❡❞❣❡s ✇✐t❤

♠❛r❦ x ∈ Ξ ❛♥❞ ✉n(t) ♠❛♥② ✈❡rt✐❝❡s ✇✐t❤ ♠❛r❦ t ∈ Θ✳ G(n)

♠n,✉n(µ, ǫ) = {G ∈ G(n) ♠n,✉n : U(G) ∈ B(µ, ǫ)}✳

❋♦r µ ∈ P(G∗) ❛♥❞ x ∈ Ξ✱ ❞❡❣x(µ)✿ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ ❡❞❣❡s ❝♦♥♥❡❝t❡❞ t♦ t❤❡ r♦♦t ✇✐t❤ ♠❛r❦ x✱ t ∈ Θ✱ Πt(µ)✿ ♣r♦❜❛❜✐❧✐t② ♦❢ r♦♦t ❤❛✈✐♥❣ ♠❛r❦ t✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✾ ✴ ✼✵

❚❤❡ ❇❈ ❡♥tr♦♣②✿ ❝♦✉♥t✐♥❣ t②♣✐❝❛❧ ❣r❛♣❤s

❋✐① s❡q✉❡♥❝❡s ♠n, ✉n s✉❝❤ t❤❛t ♠n(x)/n → ❞❡❣x(µ)/✷ ❛♥❞ ✉n(t)/n → Πt(µ) ❢♦r ❛❧❧ x ∈ Ξ, t ∈ Θ✳ ❧♦❣ |G(n)

♠n,✉n| = ♠n✶ ❧♦❣ n + cn + o(n) ✇❤❡r❡

♠n✶ =

x∈Ξ ♠n(x)✳

Σ(µ) := ❧✐♠

ǫ↓✵ ❧✐♠ s✉♣ n→∞

❧♦❣ |● (n)

♠n,✉n(µ, ǫ)| − ♠n✶ ❧♦❣ n

n Σ(µ) := ❧✐♠

ǫ↓✵ ❧✐♠ ✐♥❢ n→∞

❧♦❣ |● (n)

♠n,✉n(µ, ǫ)| − ♠n✶ ❧♦❣ n

n ■❢ t❤❡② ❛r❡ ❡q✉❛❧✱ ❞❡✜♥❡ t❤❡ ❝♦♠♠♦♥ ✈❛❧✉❡ ❛s Σ(µ) ✭●❡♥❡r❛❧✐③✐♥❣ ✇♦r❦ ♦❢ ❇♦r❞❡♥❛✈❡ ❛♥❞ ❈❛♣✉t♦✮

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✵ ✴ ✼✵

❖✉t❧✐♥❡

✶

❆ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠ st✉❞✐❡❞ ❜② ❍❛❥❡❦

✷

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❣r❛♣❤s

✸

❚❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ❧♦❝❛❧ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡

✹

▲♦❛❞ ❜❛❧❛♥❝✐♥❣ ♦♥ ❤②♣❡r❣r❛♣❤s

✺

• r❛♣❤ ✐♥❞❡①❡❞ ❞❛t❛

✻

❯♥✐✈❡rs❛❧ ❝♦♠♣r❡ss✐♦♥ ♦❢ ❣r❛♣❤✐❝❛❧ ❞❛t❛

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✶ ✴ ✼✵

❖✉r t❛r❣❡t ❢♦r t❤❡ ❣r❛♣❤ r❡❣✐♠❡

• ♦❛❧✿ ❞❡s✐❣♥ fn : Gn → {✵, ✶}∗ ❛♥❞ gn : {✵, ✶}∗ → Gn

gn ◦ fn = ■❞ µ ∈ P(G∗) ❛ ♣r♦❝❡ss ❚❛r❣❡t✿ t②♣✐❝❛❧ ❣r❛♣❤s ❖♣t✐♠❛❧ ✐❢ Gn

lwc

→ µ ❧✐♠ s✉♣

n→∞

l(fn(Gn)) − mn ❧♦❣ n n ≤ Σ(µ), ✇❤❡r❡ mn ✐s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❡❞❣❡s ✐♥ Gn✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✷ ✴ ✼✵

❆ ❋✐rst ❙t❡♣ ❈♦❞✐♥❣ ❙❝❤❡♠❡ ✿ ❊①❛♠♣❧❡

Akn,∆n = {[G, o] ∈ G∗ : ❞❡♣t❤ ≤ kn, ♠❛① ❞❡❣ ≤ ∆n} ✹✱ ✶ ✶ ✷ ✹ ✸ ✷

✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✹

t❤❡ s❡t ♦❢ ❣r❛♣❤s ✇✐t❤ t❤❡ s❛♠❡ s❡q✉❡♥❝❡

✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✸ ✴ ✼✵

❆ ❋✐rst ❙t❡♣ ❈♦❞✐♥❣ ❙❝❤❡♠❡ ✿ ❊①❛♠♣❧❡

Akn,∆n = {[G, o] ∈ G∗ : ❞❡♣t❤ ≤ kn, ♠❛① ❞❡❣ ≤ ∆n} n = ✹✱ kn = ✶ ✶ ✷ ✹ ✸ ✷

✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✹

t❤❡ s❡t ♦❢ ❣r❛♣❤s ✇✐t❤ t❤❡ s❛♠❡ s❡q✉❡♥❝❡

✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✸ ✴ ✼✵

❆ ❋✐rst ❙t❡♣ ❈♦❞✐♥❣ ❙❝❤❡♠❡ ✿ ❊①❛♠♣❧❡

Akn,∆n = {[G, o] ∈ G∗ : ❞❡♣t❤ ≤ kn, ♠❛① ❞❡❣ ≤ ∆n} n = ✹✱ kn = ✶ ✶ ✷ ✹ ✸ ∆n = ✷

✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✹

t❤❡ s❡t ♦❢ ❣r❛♣❤s ✇✐t❤ t❤❡ s❛♠❡ s❡q✉❡♥❝❡

✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✸ ✴ ✼✵

❆ ❋✐rst ❙t❡♣ ❈♦❞✐♥❣ ❙❝❤❡♠❡ ✿ ❊①❛♠♣❧❡

Akn,∆n = {[G, o] ∈ G∗ : ❞❡♣t❤ ≤ kn, ♠❛① ❞❡❣ ≤ ∆n} n = ✹✱ kn = ✶ ✶ ✷ ✹ ✸ ∆n = ✷

✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✹

t❤❡ s❡t ♦❢ ❣r❛♣❤s ✇✐t❤ t❤❡ s❛♠❡ s❡q✉❡♥❝❡

✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✸ ✴ ✼✵

❆ ❋✐rst ❙t❡♣ ❈♦❞✐♥❣ ❙❝❤❡♠❡ ✿ ❊①❛♠♣❧❡

Akn,∆n = {[G, o] ∈ G∗ : ❞❡♣t❤ ≤ kn, ♠❛① ❞❡❣ ≤ ∆n} n = ✹✱ kn = ✶ ✶ ✷ ✹ ✸ ∆n = ✷

✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✹

t❤❡ s❡t ♦❢ ❣r❛♣❤s ✇✐t❤ t❤❡ s❛♠❡ s❡q✉❡♥❝❡

✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✸ ✴ ✼✵

❆ ❋✐rst ❙t❡♣ ❈♦❞✐♥❣ ❙❝❤❡♠❡ ✿ ❊①❛♠♣❧❡

Akn,∆n = {[G, o] ∈ G∗ : ❞❡♣t❤ ≤ kn, ♠❛① ❞❡❣ ≤ ∆n} n = ✹✱ kn = ✶ ✶ ✷ ✹ ✸ ∆n = ✷

✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✹

Wn := t❤❡ s❡t ♦❢ ❣r❛♣❤s ✇✐t❤ t❤❡ s❛♠❡ s❡q✉❡♥❝❡

✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✸ ✴ ✼✵

❆ ❋✐rst ❙t❡♣ ❈♦❞✐♥❣ ❙❝❤❡♠❡ ✿ ❊①❛♠♣❧❡

Akn,∆n = {[G, o] ∈ G∗ : ❞❡♣t❤ ≤ kn, ♠❛① ❞❡❣ ≤ ∆n} n = ✹✱ kn = ✶ ✶ ✷ ✹ ✸ ∆n = ✷

✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✹

Wn := t❤❡ s❡t ♦❢ ❣r❛♣❤s ✇✐t❤ t❤❡ s❛♠❡ s❡q✉❡♥❝❡

✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸ ✶ ✷ ✹ ✸

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✸ ✴ ✼✵

❆♥❛❧②s✐s ❖✉t❧✐♥❡

l(fn(Gn))✱ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❜✐ts ✇❡ ✉s❡✿

◮ ❧♦❣ n ❜✐ts ❢♦r ∆n✱ ◮ |Akn,∆n| ❧♦❣ n ❜✐ts ❢♦r s♣❡❝✐❢②✐♥❣ ❤♦✇ ♠❛♥② t✐♠❡s ❡❛❝❤ ♣❛tt❡r♥

❛♣♣❡❛rs ✐♥ t❤❡ ❣r❛♣❤

◮ ❧♦❣ |Wn| ❜✐ts t♦ s♣❡❝✐❢② t❤❡ ✐♥♣✉t ❣r❛♣❤ ❛♠♦♥❣ t❤❡ ❣r❛♣❤s ✇✐t❤

t❤❡ s❛♠❡ ♣❛tt❡r♥ ❝♦✉♥ts✳

❲❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t ✐❢ Gn

lwc

→ µ✱ l(fn(Gn)) − mn ❧♦❣ n n ≤ Σ(µ). ■❢ |Akn,∆n| = o(n/ ❧♦❣ n)✱ ✇❡ ♦♥❧② ♥❡❡❞ t♦ ❝♦♥s✐❞❡r t❤❡ ❧♦❣ |Wn| t❡r♠✳

• r❛♣❤s ✐♥ Wn ❛r❡ t②♣✐❝❛❧ ⇒ ②✐❡❧❞s Σ(µ) ❛s ❛♥ ✉♣♣❡r ❜♦✉♥❞✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✹ ✴ ✼✵

❋✐rst st❡♣ ❛❧❣♦r✐t❤♠✿ ▼❛✐♥ ❘❡s✉❧t

Pr♦♣♦s✐t✐♦♥

■❢ ♣❛r❛♠❡t❡rs kn ❛♥❞ ∆n ❛r❡ s✉❝❤ t❤❛t |Akn,∆n| = o(

n ❧♦❣ n) ❛♥❞

kn → ∞ ❛s n → ∞✱ ❢♦r ❛♥② s❡q✉❡♥❝❡ Gn ✇✐t❤ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ♥♦ ♠♦r❡ t❤❛♥ ∆n ❛♥❞ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t µ ∈ P(G∗) s✉❝❤ t❤❛t Σ(µ) > −∞ ✇❡ ❤❛✈❡ ❧✐♠ s✉♣

n→∞

l(fn(Gn)) − mn ❧♦❣ n n ≤ Σ(µ), ✭✶✮ ✇❤❡r❡ mn ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s ✐♥ Gn✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✺ ✴ ✼✵

• ❡♥❡r❛❧ ❆❧❣♦r✐t❤♠

Gn

✺

❡♥❞♣♦✐♥t ♦❢ r❡♠♦✈❡❞ ❡❞❣❡s

❈♦♠♣r❡ss ✉s✐♥❣ t❤❡ ✜rst st❡♣ s❝❤❡♠❡✱ t❤❡♥ ❝♦♠♣r❡ss r❡♠♦✈❡❞ ❡❞❣❡s ❧♦❣ ❧♦❣ ❧♦❣ ❧♦❣ ✵ ❧♦❣

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✻ ✴ ✼✵

• ❡♥❡r❛❧ ❆❧❣♦r✐t❤♠

Gn

∆n=✺

→

❡♥❞♣♦✐♥t ♦❢ r❡♠♦✈❡❞ ❡❞❣❡s

❈♦♠♣r❡ss ✉s✐♥❣ t❤❡ ✜rst st❡♣ s❝❤❡♠❡✱ t❤❡♥ ❝♦♠♣r❡ss r❡♠♦✈❡❞ ❡❞❣❡s ❧♦❣ ❧♦❣ ❧♦❣ ❧♦❣ ✵ ❧♦❣

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✻ ✴ ✼✵

• ❡♥❡r❛❧ ❆❧❣♦r✐t❤♠

Gn

∆n=✺

→ ˜ Gn

❡♥❞♣♦✐♥t ♦❢ r❡♠♦✈❡❞ ❡❞❣❡s

❈♦♠♣r❡ss ✉s✐♥❣ t❤❡ ✜rst st❡♣ s❝❤❡♠❡✱ t❤❡♥ ❝♦♠♣r❡ss r❡♠♦✈❡❞ ❡❞❣❡s ❧♦❣ ❧♦❣ ❧♦❣ ❧♦❣ ✵ ❧♦❣

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✻ ✴ ✼✵

• ❡♥❡r❛❧ ❆❧❣♦r✐t❤♠

Gn

∆n=✺

→ ˜ Gn

Tn = {❡♥❞♣♦✐♥t ♦❢ r❡♠♦✈❡❞ ❡❞❣❡s}

❈♦♠♣r❡ss ✉s✐♥❣ t❤❡ ✜rst st❡♣ s❝❤❡♠❡✱ t❤❡♥ ❝♦♠♣r❡ss r❡♠♦✈❡❞ ❡❞❣❡s ❧♦❣ ❧♦❣ ❧♦❣ ❧♦❣ ✵ ❧♦❣

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✻ ✴ ✼✵

• ❡♥❡r❛❧ ❆❧❣♦r✐t❤♠

Gn

∆n=✺

→ ˜ Gn

Tn = {❡♥❞♣♦✐♥t ♦❢ r❡♠♦✈❡❞ ❡❞❣❡s}

❈♦♠♣r❡ss ˜ Gn ✉s✐♥❣ t❤❡ ✜rst st❡♣ s❝❤❡♠❡✱ t❤❡♥ ❝♦♠♣r❡ss r❡♠♦✈❡❞ ❡❞❣❡s ❧♦❣ ❧♦❣ ❧♦❣ ❧♦❣ ✵ ❧♦❣

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✻ ✴ ✼✵

• ❡♥❡r❛❧ ❆❧❣♦r✐t❤♠

Gn

∆n=✺

→ ˜ Gn

Tn = {❡♥❞♣♦✐♥t ♦❢ r❡♠♦✈❡❞ ❡❞❣❡s}

❈♦♠♣r❡ss ˜ Gn ✉s✐♥❣ t❤❡ ✜rst st❡♣ s❝❤❡♠❡✱ t❤❡♥ ❝♦♠♣r❡ss r❡♠♦✈❡❞ ❡❞❣❡s ∆n = ❧♦❣ ❧♦❣ n kn =

• ❧♦❣ ❧♦❣ n

✵ ❧♦❣

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✻ ✴ ✼✵

• ❡♥❡r❛❧ ❆❧❣♦r✐t❤♠

Gn

∆n=✺

→ ˜ Gn

Tn = {❡♥❞♣♦✐♥t ♦❢ r❡♠♦✈❡❞ ❡❞❣❡s}

❈♦♠♣r❡ss ˜ Gn ✉s✐♥❣ t❤❡ ✜rst st❡♣ s❝❤❡♠❡✱ t❤❡♥ ❝♦♠♣r❡ss r❡♠♦✈❡❞ ❡❞❣❡s ∆n = ❧♦❣ ❧♦❣ n kn =

• ❧♦❣ ❧♦❣ n

|Tn|/n → ✵ |Akn,∆n| = o(n/ ❧♦❣ n) Gn

lwc

→ µ ⇒ ˜ Gn

lwc

→ µ

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✻ ✴ ✼✵

❘❡s✉❧t✿ ❆❝❤✐❡✈❛❜✐❧✐t②

❚❤❡♦r❡♠

❆ss✉♠❡ µ ∈ G∗ ✇✐t❤ ❞❡❣x(µ) < ∞ ❢♦r ❛❧❧ x ❛♥❞ Σ(µ) > −∞✳ ■❢ Gn ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♠❛r❦❡❞ ❣r❛♣❤s ✇✐t❤ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t µ✱ ✇❡ ❤❛✈❡ ❧✐♠ s✉♣

n→∞

l(fn(Gn)) − mn ❧♦❣ n n ≤ Σ(µ), ✇❤❡r❡ mn ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s ✐♥ Gn✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✼ ✴ ✼✵

❘❡s✉❧t✿ ❈♦♥✈❡rs❡

❚❤❡♦r❡♠

❆ss✉♠❡ µ ∈ P(G∗) ✇✐t❤ Σ(µ) > −∞ ❛♥❞ ❞❡❣x(µ) < ∞ ❢♦r ❛❧❧ x ∈ Ξ✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ s❡q✉❡♥❝❡ ♦❢ ❣r❛♣❤ ❡♥s❡♠❜❧❡s Gn ❝♦♥✈❡r❣✐♥❣ t♦ µ s✉❝❤ t❤❛t ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♦♥❡ ❢♦r ❛♥② s❡q✉❡♥❝❡ ♦❢ ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s fn ✇❡ ❤❛✈❡ ❧✐♠ ✐♥❢

n→∞

l(fn(Gn)) − mn ❧♦❣ n n ≥ Σ(µ), ✇❤❡r❡ mn ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s ✐♥ Gn✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✽ ✴ ✼✵

❈♦♥❝✉❞✐♥❣ r❡♠❛r❦s

❏✉st ❧✐❦❡ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✐s ❛ ♠♦❞❡❧ ❢♦r t❤❡ st❛t✐st✐❝s ♦❢ ❛ ❧♦♥❣ str✐♥❣ ♦❢ ❞❛t❛✱ ❛ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t ♦❢ ♠❛r❦❡❞ ❣r❛♣❤s ✐s ❛ ♠♦❞❡❧ ❢♦r t❤❡ st❛t✐st✐❝s ♦❢ ❞❛t❛ t❤❛t ❧✐✈❡s ♦♥ ❧❛r❣❡ ❣r❛♣❤s✳ ❚❤✐s ♣r♦✈✐❞❡s ❛ ♠❡t❤♦❞♦❧♦❣② t♦ ❛❞❞r❡ss ♥❡t✇♦r❦✐♥❣ ♣r♦❜❧❡♠s ❛♥❞ ❞❛t❛ ❝❡♥tr✐❝ ♣r♦❜❧❡♠s ❛r✐s✐♥❣ ✐♥ ♥❡t✇♦r❦s t❤❛t ♣❛r❛❧❧❡❧s ❤♦✇ st♦❝❤❛st✐❝ ♣r♦❝❡sss ❛r❡ ✉s❡❞ ✐♥ t❤❡ st✉❞② ♦❢ t✐♠❡ s❡r✐❡s✳ ❚❤✐s ✇❛s ✐❧❧✉str❛t❡❞ ✇✐t❤ t✇♦ ❦✐♥❞s ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✿ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ✐♥ ❣r❛♣❤s ❛♥❞ ❤②♣❡r❣r❛♣❤s✱ ❛♥❞ ✉♥✐✈❡rs❛❧ ❧♦ss❧❡sss ❝♦♠♣r❡ss✐♦♥ ♦❢ ❣r❛♣❤✲str✉❝t✉r❡❞ ❞❛t❛✳ ❆ ✇♦r❧❞ ♦❢ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s ❛✇❛✐ts✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✾ ✴ ✼✵

❈♦♥❝✉❞✐♥❣ r❡♠❛r❦s

❏✉st ❧✐❦❡ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✐s ❛ ♠♦❞❡❧ ❢♦r t❤❡ st❛t✐st✐❝s ♦❢ ❛ ❧♦♥❣ str✐♥❣ ♦❢ ❞❛t❛✱ ❛ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t ♦❢ ♠❛r❦❡❞ ❣r❛♣❤s ✐s ❛ ♠♦❞❡❧ ❢♦r t❤❡ st❛t✐st✐❝s ♦❢ ❞❛t❛ t❤❛t ❧✐✈❡s ♦♥ ❧❛r❣❡ ❣r❛♣❤s✳ ❚❤✐s ♣r♦✈✐❞❡s ❛ ♠❡t❤♦❞♦❧♦❣② t♦ ❛❞❞r❡ss ♥❡t✇♦r❦✐♥❣ ♣r♦❜❧❡♠s ❛♥❞ ❞❛t❛ ❝❡♥tr✐❝ ♣r♦❜❧❡♠s ❛r✐s✐♥❣ ✐♥ ♥❡t✇♦r❦s t❤❛t ♣❛r❛❧❧❡❧s ❤♦✇ st♦❝❤❛st✐❝ ♣r♦❝❡sss ❛r❡ ✉s❡❞ ✐♥ t❤❡ st✉❞② ♦❢ t✐♠❡ s❡r✐❡s✳ ❚❤✐s ✇❛s ✐❧❧✉str❛t❡❞ ✇✐t❤ t✇♦ ❦✐♥❞s ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✿ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ✐♥ ❣r❛♣❤s ❛♥❞ ❤②♣❡r❣r❛♣❤s✱ ❛♥❞ ✉♥✐✈❡rs❛❧ ❧♦ss❧❡sss ❝♦♠♣r❡ss✐♦♥ ♦❢ ❣r❛♣❤✲str✉❝t✉r❡❞ ❞❛t❛✳ ❆ ✇♦r❧❞ ♦❢ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s ❛✇❛✐ts✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✾ ✴ ✼✵

❈♦♥❝✉❞✐♥❣ r❡♠❛r❦s

❏✉st ❧✐❦❡ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✐s ❛ ♠♦❞❡❧ ❢♦r t❤❡ st❛t✐st✐❝s ♦❢ ❛ ❧♦♥❣ str✐♥❣ ♦❢ ❞❛t❛✱ ❛ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t ♦❢ ♠❛r❦❡❞ ❣r❛♣❤s ✐s ❛ ♠♦❞❡❧ ❢♦r t❤❡ st❛t✐st✐❝s ♦❢ ❞❛t❛ t❤❛t ❧✐✈❡s ♦♥ ❧❛r❣❡ ❣r❛♣❤s✳ ❚❤✐s ♣r♦✈✐❞❡s ❛ ♠❡t❤♦❞♦❧♦❣② t♦ ❛❞❞r❡ss ♥❡t✇♦r❦✐♥❣ ♣r♦❜❧❡♠s ❛♥❞ ❞❛t❛ ❝❡♥tr✐❝ ♣r♦❜❧❡♠s ❛r✐s✐♥❣ ✐♥ ♥❡t✇♦r❦s t❤❛t ♣❛r❛❧❧❡❧s ❤♦✇ st♦❝❤❛st✐❝ ♣r♦❝❡sss ❛r❡ ✉s❡❞ ✐♥ t❤❡ st✉❞② ♦❢ t✐♠❡ s❡r✐❡s✳ ❚❤✐s ✇❛s ✐❧❧✉str❛t❡❞ ✇✐t❤ t✇♦ ❦✐♥❞s ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✿ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ✐♥ ❣r❛♣❤s ❛♥❞ ❤②♣❡r❣r❛♣❤s✱ ❛♥❞ ✉♥✐✈❡rs❛❧ ❧♦ss❧❡sss ❝♦♠♣r❡ss✐♦♥ ♦❢ ❣r❛♣❤✲str✉❝t✉r❡❞ ❞❛t❛✳ ❆ ✇♦r❧❞ ♦❢ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s ❛✇❛✐ts✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✾ ✴ ✼✵

❈♦♥❝✉❞✐♥❣ r❡♠❛r❦s

❏✉st ❧✐❦❡ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ✐s ❛ ♠♦❞❡❧ ❢♦r t❤❡ st❛t✐st✐❝s ♦❢ ❛ ❧♦♥❣ str✐♥❣ ♦❢ ❞❛t❛✱ ❛ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐t ♦❢ ♠❛r❦❡❞ ❣r❛♣❤s ✐s ❛ ♠♦❞❡❧ ❢♦r t❤❡ st❛t✐st✐❝s ♦❢ ❞❛t❛ t❤❛t ❧✐✈❡s ♦♥ ❧❛r❣❡ ❣r❛♣❤s✳ ❚❤✐s ♣r♦✈✐❞❡s ❛ ♠❡t❤♦❞♦❧♦❣② t♦ ❛❞❞r❡ss ♥❡t✇♦r❦✐♥❣ ♣r♦❜❧❡♠s ❛♥❞ ❞❛t❛ ❝❡♥tr✐❝ ♣r♦❜❧❡♠s ❛r✐s✐♥❣ ✐♥ ♥❡t✇♦r❦s t❤❛t ♣❛r❛❧❧❡❧s ❤♦✇ st♦❝❤❛st✐❝ ♣r♦❝❡sss ❛r❡ ✉s❡❞ ✐♥ t❤❡ st✉❞② ♦❢ t✐♠❡ s❡r✐❡s✳ ❚❤✐s ✇❛s ✐❧❧✉str❛t❡❞ ✇✐t❤ t✇♦ ❦✐♥❞s ♦❢ ❛♣♣❧✐❝❛t✐♦♥s✿ r❡s♦✉r❝❡ ❛❧❧♦❝❛t✐♦♥ ✐♥ ❣r❛♣❤s ❛♥❞ ❤②♣❡r❣r❛♣❤s✱ ❛♥❞ ✉♥✐✈❡rs❛❧ ❧♦ss❧❡sss ❝♦♠♣r❡ss✐♦♥ ♦❢ ❣r❛♣❤✲str✉❝t✉r❡❞ ❞❛t❛✳ ❆ ✇♦r❧❞ ♦❢ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s ❛✇❛✐ts✳

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✻✾ ✴ ✼✵

❚❤❡ ❊♥❞

❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✼✵ ✴ ✼✵