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▲♦❛❞ ♣❡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❝❤ ✶✵✱ ✷✵✶✼ ✶✶ ✴ ✼✵

■♥ 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 ✐♥❞✉❝❡❞ ❧♦❛❞ ✵✳ ▲♦❛❞ ♣❡r❝♦❧❛t✐♦♥ ❛s ♥♦♥✉♥✐q✉❡♥❡ss ✐♥ t❤❡ ❧✐♠✐t ❆♥ ✐♥✜♥✐t❡ s♣❛rs❡ ❣r❛♣❤ ❝❛♥ ❡①❤✐❜✐t ♥♦♥✉♥✐q✉❡♥❡ss ✐♥ ✐ts ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥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❝❤ ✶✵✱ ✷✵✶✼ ✶✷ ✴ ✼✵

◆♦✇ ✢✐♣ t❤❡ ❛❧❧♦❝❛t✐♦♥ ♦❢ ❡❛❝❤ ❡❞❣❡✳ ❚❤✐s ✐s ❛♥♦t❤❡r ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ ✦✦ ✳ ❚❤❡ ✐♥❞✉❝❡❞ ❧♦❛❞ ❛t ❡❛❝❤ ✈❡rt❡① ✐s ✷✳ ❚❤❡s❡ ❡①❛♠♣❧❡s ❛r❡ ❞✉❡ t♦ ❍❛❥❡❦✳ ◆♦♥✉♥✐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❤❛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 ✭ ❍❛❥❡❦ ✮ α M ❝♦♥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 P♦✐ ✷ ✸ ✶ ✷ ▲❛r❣❡ ❊r❞➤s ❘é♥②✐ ❣r❛♣❤s G ( n , α/ n ) ❱❡♥❦❛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❝❤ ✶✵✱ ✷✵✶✼ ✶✾ ✴ ✼✵

❚❤❡ ❧♦❝❛❧ ❡♥✈✐r♦♥♠❡♥t ♦❢ ❛ t②♣✐❝❛❧ ✈❡rt❡① ✐♥ ❛♥ ❊r❞➤s ✲ ❘é♥②✐ ❣r❛♣❤ ❝♦♥✈❡r❣❡s t♦ ❛ P♦✐ss♦♥ ●❛❧t♦♥✲❲❛ts♦♥ tr❡❡ ❛s ✳ ❚❤❡ 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❝✳ ❱❡♥❦❛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 + ✶ ✶ � δ x i ,..., x i + L − ✶ ⇒ P X ✵ ,..., X L − ✶ . ✷ ( N + ✶ ) − L i = − N ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✺ ✴ ✼✵

✏❊♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥✑ ♦❢ ❛ ♠❛r❦❡❞ ❣r❛♣❤ ✶ ✶ ✶ ✷ ✹ ✷ ✸ ✹ ✺ ✶ ✹ ✻ ✼ G U ✷ ( G ) ✽ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✻ ✴ ✼✵

✿ s♣❛❝❡ ♦❢ ✉♥❧❛❜❡❧❧❡❞ ♠❛r❦❡❞ r♦♦t❡❞ ❣r❛♣❤s ❆ ♣r♦❝❡ss ✇✐t❤ ✈❛❧✉❡s ✐♥ r♦♦t❡❞ ♠❛r❦❡❞ ❣r❛♣❤s✿ ❲❡ ✇✐❧❧ ✜rst ❝♦♥s✐❞❡r t❤❡ ✉♥♠❛r❦❡❞ ❝❛s❡✳ ❘♦♦t❡❞ ♠❛r❦❡❞ ❣r❛♣❤ ♣r♦❝❡ss ❢r♦♠ ❛ ♠❛r❦❡❞ ❣r❛♣❤ ✶ ✶ ✷ ✹ ✶ ✷ ✸ ✹ ✶ ✹ ✺ ✻ ✼ G ✽ U ( G ) ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✷✼ ✴ ✼✵

❆ ♣r♦❝❡ss ✇✐t❤ ✈❛❧✉❡s ✐♥ r♦♦t❡❞ ♠❛r❦❡❞ ❣r❛♣❤s✿ ❲❡ ✇✐❧❧ ✜rst ❝♦♥s✐❞❡r t❤❡ ✉♥♠❛r❦❡❞ ❝❛s❡✳ ❘♦♦t❡❞ ♠❛r❦❡❞ ❣r❛♣❤ ♣r♦❝❡ss ❢r♦♠ ❛ ♠❛r❦❡❞ ❣r❛♣❤ ✶ ✶ ✷ ✹ ✶ ✷ ✸ ✹ ✶ ✹ ✺ ✻ ✼ G ✽ U ( G ) G ∗ ✿ s♣❛❝❡ ♦❢ ✉♥❧❛❜❡❧❧❡❞ ♠❛r❦❡❞ 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 ) = f ( G , i , o ) . i ∼ o ❆ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ µ ♦♥ G ∗ ❣✐✈❡s r✐s❡ t♦ ❛ ♠❡❛s✉r❡ � µ ♦♥ G ∗∗ ✈✐❛ � � ❢♦r ❛❧❧ ❜♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s f . fd � µ = ∂ fd µ , G ∗∗ G ∗ � ◆♦t❡ t❤❛t � µ ( G ∗∗ ) = ❞❡❣ ( µ ) := G ∗ ❞❡❣ ( r♦♦t ) d µ ✳ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✸✶ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r✐t② : G ∗∗ �→ R ✱ ❞❡✜♥❡ f ∗ : G ∗∗ �→ R ✈✐❛ ●✐✈❡♥ f f ∗ ( G , i , o ) = f ( G , o , i ) . ❆ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ µ ♦♥ G ∗ ✐s ❝❛❧❧❡❞ ✉♥✐♠♦❞✉❧❛r ✐❢ � � f ∗ d � fd � µ = µ , ❢♦r ❛❧❧ ❜♦✉♥❞❡❞ ❝♦♥t✐♥✉♦✉s f . G ∗∗ G ∗∗ ■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) d µ ♦✈❡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 ≥ ✵✱ � ( ∂ Θ ✵ − t ) + d µ . Φ µ ( t ) := G ∗ t �→ Φ µ ( 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✐♦♥ { ✶ � � ˆ Φ µ ( t ) = ♠❛① f d � µ − t fd µ } , ✷ f : G ∗ → [ ✵ , ✶ ] , ❇♦r❡❧ G ∗∗ G ∗ ❢♦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 ✶ � ✶ � ˆ ( ∂ ˆ f d � µ = f ) d µ ✷ ✷ G ∗∗ G ∗ ✶ � � = ✶ ( ∂ Θ ✵ ( G , i ) > t ❛♥❞ ∂ Θ ✵ ( G , o ) > t ) d µ ✷ G ∗ i ∼ o ❚❤✉s � ( ∂ Θ ✵ − t ) + d µ = ✶ � � ˆ f d � µ − t fd µ , ✷ G ∗ G ∗∗ G ∗ ❢♦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 i π ( i ) ✱ ❞❡✜♥❡ π ( i ) := ( i + ✶ ) π ( i + ✶ ) ˆ , i ≥ ✵ . � i i π ( 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 ✱ ✇❡ ❤❛✈❡ Q = F π, t ( Q ) { E [ D ] Φ µ ( t ) = ♠❛① P ( ξ ✶ + ξ ✷ > ✶ ) − tP ( ξ ✶ + . . . + ξ D > t ) } , ✷ D ] ✶ ✇❤❡r❡ F π, t ( Q ) ✐s t❤❡ ❧❛✇ ♦❢ [ ✶ − t + ξ ✶ + . . . + ξ ˆ ✵ ✳ ✵ ❡q✉❛❧s ✵ ✐❢ a < ✵✱ ✶ ✐❢ a > ✶ ❛♥❞ a ♦t❤❡r✇✐s❡✳ ❆❧s♦✱ ˆ ❍❡r❡ [ a ] ✶ D ❤❛s t❤❡ ❧❛✇ ˆ π ✱ D ❤❛s t❤❡ ❧❛✇ π ✱ ❛♥❞ t❤❡ ξ i ❛r❡ ✐✳✐✳❞✳ ✇✐t❤ ❧❛✇ Q ✳ ❘❡❝❛❧❧ t❤❛t � ( ∂ Θ ✵ − t ) + d µ , t �→ Φ µ ( t ) := G ∗ ❝❤❛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 = F π, t ( Q ) ✱ ✇❤❡r❡ F π, t ( Q ) ✐s t❤❡ ❧❛✇ ♦❢ D ] ✶ [ ✶ − t + ξ ✶ + . . . + ξ ˆ ✵ ✱ ✇❤❡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 ✱ ✶ − ξ k ❤❛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 ✳ D )] ✶ ❚❤❡ ♥✉♠❜❡r [ ✶ − ( t − ξ ✶ − . . . − ξ ˆ ✵ ✐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 > ✵ ✇❡ ❤❛✈❡ n { ✶ � e λ d n ( i ) } < ∞ . s✉♣ n n ≥ ✶ i = ✶ ▲❡t Z ( n ) δ, t ❞❡♥♦t❡ t❤❡ ♥✉♠❜❡r ♦❢ s✉❜s❡ts S ♦❢ { ✶ , . . . , n } ♦❢ s✐③❡ | S | ≤ δ n ✇✐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 ✶ ✶ / ✷ ∂θ ( ✶ ) = ✶ e ✶ v ✶ v ✷ ∂θ ( ✷ ) = ✶ ✷ ✵ θ ( e ✶ , v ✶ ) = ✶ / ✷ ✵ v ✷ ✶ e ✶ ✶ ✶ / ✷ e ✷ e ✷ ✶ / ✷ v ✸ ✵ ✵ ✵ ✵ ✶ ∂θ ( ✸ ) = ✶ e ✸ v ✸ e ✸ v ✹ ∂θ ( ✹ ) = ✶ ✷ ✶ v ✹ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✸ ✴ ✼✵

❙✐♠♣❧❡✱ ❝♦♥♥❡❝t❡❞✱ ✜♥✐t❡ ❡❞❣❡s✱ ❧♦❝❛❧❧② ✜♥✐t❡ H ∗ ❛♥❞ H ∗∗ i H ∗ = { [ H , i ] } ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✹ ✴ ✼✵

❙✐♠♣❧❡✱ ❝♦♥♥❡❝t❡❞✱ ✜♥✐t❡ ❡❞❣❡s✱ ❧♦❝❛❧❧② ✜♥✐t❡ H ∗ ❛♥❞ H ∗∗ i i e H ∗ = { [ H , i ] } H ∗∗ = { [ H , e , i ] } ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✹ ✴ ✼✵

H ∗ ❛♥❞ H ∗∗ i i e H ∗ = { [ H , i ] } H ∗∗ = { [ H , e , i ] } ❙✐♠♣❧❡✱ ❝♦♥♥❡❝t❡❞✱ ✜♥✐t❡ ❡❞❣❡s✱ ❧♦❝❛❧❧② ✜♥✐t❡ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✹ ✴ ✼✵

✇❤❡♥ ◆♦t ❛❧❧ ❝❛♥ ❜❡ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐ts ♦❢ ✜♥✐t❡ ❤②♣❡r❣r❛♣❤s ❋♦r ✱ ❧❡t ❯♥✐♠♦❞✉❧❛r✐t② ❋✐♥✐t❡ H n ✶ � U ( H ) = δ [ H , i ] ∈ P ( H ∗ ) | V ( H ) | i ∈ V ( H ) ❯ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✺ ✴ ✼✵

◆♦t ❛❧❧ ❝❛♥ ❜❡ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐ts ♦❢ ✜♥✐t❡ ❤②♣❡r❣r❛♣❤s ❋♦r ✱ ❧❡t ❯♥✐♠♦❞✉❧❛r✐t② ❋✐♥✐t❡ H n ✶ � U ( H ) = δ [ H , i ] ∈ P ( H ∗ ) | V ( H ) | i ∈ V ( H ) lwc H n → µ ✇❤❡♥ U ( H n ) ⇒ µ ❯ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✺ ✴ ✼✵

❋♦r ✱ ❧❡t ❯♥✐♠♦❞✉❧❛r✐t② ❋✐♥✐t❡ H n ✶ � U ( H ) = δ [ H , i ] ∈ P ( H ∗ ) | V ( H ) | i ∈ V ( H ) lwc H n → µ ✇❤❡♥ U ( H n ) ⇒ µ ◆♦t ❛❧❧ µ ❝❛♥ ❜❡ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐ts ♦❢ ✜♥✐t❡ ❤②♣❡r❣r❛♣❤s ❯ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✺ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r✐t② ❋✐♥✐t❡ H n ✶ � U ( H ) = δ [ H , i ] ∈ P ( H ∗ ) | V ( H ) | i ∈ V ( H ) lwc H n → µ ✇❤❡♥ U ( H n ) ⇒ µ ◆♦t ❛❧❧ µ ❝❛♥ ❜❡ ❧♦❝❛❧ ✇❡❛❦ ❧✐♠✐ts ♦❢ ✜♥✐t❡ ❤②♣❡r❣r❛♣❤s ❋♦r f : H ∗∗ → R ✱ ❧❡t � ∂ f : H ∗ → R ∂ f ( H , i ) = f ( H , e , i ) e ∋ i ❯ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✺ ✴ ✼✵

❋♦r ✱ ❧❡t ✶ ✐s ❝❛❧❧❡❞ ✉♥✐♠♦❞✉❧❛r ✐❢ ■❢ ✱ ✐s ✉♥✐♠♦❞✉❧❛r ❯♥✐♠♦❞✉❧❛r✐t② ✭❝♦♥t✬❞✮ ❋♦r µ ∈ P ( H ∗ ) ✱ ❞❡✜♥❡ � µ ∈ M ( H ∗∗ ) ❛s � � fd � µ = ∂ fd µ ❢♦r ❛❧❧ ❇♦r❡❧ ❢✉♥❝t✐♦♥ f ♦♥ H ∗∗ ✳ ❯ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✻ ✴ ✼✵

✐s ❝❛❧❧❡❞ ✉♥✐♠♦❞✉❧❛r ✐❢ ■❢ ✱ ✐s ✉♥✐♠♦❞✉❧❛r ❯♥✐♠♦❞✉❧❛r✐t② ✭❝♦♥t✬❞✮ ❋♦r µ ∈ P ( H ∗ ) ✱ ❞❡✜♥❡ � µ ∈ M ( H ∗∗ ) ❛s � � fd � µ = ∂ fd µ ❢♦r ❛❧❧ ❇♦r❡❧ ❢✉♥❝t✐♦♥ f ♦♥ H ∗∗ ✳ ❋♦r f : H ∗∗ → R ✱ ❧❡t ∇ f ( H , e , i ) = ✶ � ∇ f : H ∗∗ → R f ( H , e , j ) . | e | j ∈ e ❯ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✻ ✴ ✼✵

■❢ ✱ ✐s ✉♥✐♠♦❞✉❧❛r ❯♥✐♠♦❞✉❧❛r✐t② ✭❝♦♥t✬❞✮ ❋♦r µ ∈ P ( H ∗ ) ✱ ❞❡✜♥❡ � µ ∈ M ( H ∗∗ ) ❛s � � fd � µ = ∂ fd µ ❢♦r ❛❧❧ ❇♦r❡❧ ❢✉♥❝t✐♦♥ f ♦♥ H ∗∗ ✳ ❋♦r f : H ∗∗ → R ✱ ❧❡t ∇ f ( H , e , i ) = ✶ � ∇ f : H ∗∗ → R f ( H , e , j ) . | e | j ∈ e µ ∈ P ( H ∗ ) ✐s ❝❛❧❧❡❞ ✉♥✐♠♦❞✉❧❛r ✐❢ � � fd � µ = ∇ fd � µ ❯ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✻ ✴ ✼✵

❯♥✐♠♦❞✉❧❛r✐t② ✭❝♦♥t✬❞✮ ❋♦r µ ∈ P ( H ∗ ) ✱ ❞❡✜♥❡ � µ ∈ M ( H ∗∗ ) ❛s � � fd � µ = ∂ fd µ ❢♦r ❛❧❧ ❇♦r❡❧ ❢✉♥❝t✐♦♥ f ♦♥ H ∗∗ ✳ ❋♦r f : H ∗∗ → R ✱ ❧❡t ∇ f ( H , e , i ) = ✶ � ∇ f : H ∗∗ → R f ( H , e , j ) . | e | j ∈ e µ ∈ P ( H ∗ ) ✐s ❝❛❧❧❡❞ ✉♥✐♠♦❞✉❧❛r ✐❢ � � fd � µ = ∇ fd � µ lwc ■❢ H n → µ ✱ µ ✐s ✉♥✐♠♦❞✉❧❛r ❯ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✻ ✴ ✼✵

✐s ❜❛❧❛♥❝❡❞ ✇✳r✳t✳ ✐❢ ❢♦r ✕❛❧♠♦st ❛❧❧ ✵ ❇♦r❡❧ ❆❧❧♦❝❛t✐♦♥s ❛♥❞ ❇❛❧❛♥❝❡❞♥❡ss Θ : H ∗∗ → [ ✵ , ✶ ] ✐s ❝❛❧❧❡❞ ❛ ❇♦r❡❧ ❛❧❧♦❝❛t✐♦♥ ✐❢ � Θ( H , e , j ) = ✶ ∀ [ H , e , i ] ∈ H ∗∗ j ∈ e ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✼ ✴ ✼✵

❇♦r❡❧ ❆❧❧♦❝❛t✐♦♥s ❛♥❞ ❇❛❧❛♥❝❡❞♥❡ss Θ : H ∗∗ → [ ✵ , ✶ ] ✐s ❝❛❧❧❡❞ ❛ ❇♦r❡❧ ❛❧❧♦❝❛t✐♦♥ ✐❢ � Θ( H , e , j ) = ✶ ∀ [ H , e , i ] ∈ H ∗∗ j ∈ e 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❝❤ ✶✵✱ ✷✵✶✼ ✹✼ ✴ ✼✵

✭❡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❡ ✳ ♠✐♥ ▼❛✐♥ r❡s✉❧ts ✭✇✐t❤ P❛②❛♠ ❉❡❧❣♦s❤❛✮ ❚❤❡♦r❡♠ ❚❛❦❡ µ ∈ P ( H ∗ ) ✉♥✐♠♦❞✉❧❛r✱ ❞❡❣ ( µ ) , ❱❛r ( µ ) < ∞ ✱ t❤❡♥ ❍❛❥❡❦ ❈❊ ❱❈ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✽ ✴ ✼✵

✭✉♥✐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❡ ✳ ♠✐♥ ▼❛✐♥ r❡s✉❧ts ✭✇✐t❤ P❛②❛♠ ❉❡❧❣♦s❤❛✮ ❚❤❡♦r❡♠ ❚❛❦❡ µ ∈ P ( H ∗ ) ✉♥✐♠♦❞✉❧❛r✱ ❞❡❣ ( µ ) , ❱❛r ( µ ) < ∞ ✱ t❤❡♥ ✭❡①✐st❡♥❝❡✮ ∃ ❛ ❜❛❧❛♥❝❡❞ ❛❧❧♦❝❛t✐♦♥ Θ ✵ ✶ ❍❛❥❡❦ ❈❊ ❱❈ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✽ ✴ ✼✵

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

✭♦♣t✐♠❛❧✐t②✮ ✐s ❜❛❧❛♥❝❡❞ ✐✛ ✐t ♠✐♥✐♠✐③❡s ❢♦r str✐❝t❧② ❝♦♥✈❡① ✹ ✵ ✳ ✭✈❛r✐❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❛♥❞ ❜❛❧❛♥❝❡❞✱ t❤❡♥ ✺ ♠❛① ♠✐♥ ❇♦r❡❧ ✵ ✶ ✶ ♠✐♥ ✇❤❡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✳ ✷ lwc ✭❝♦♥t✐♥✉✐t②✮ H n → µ t❤❡♥ L n ⇒ L ✸ ❍❛❥❡❦ ❈❊ ❱❈ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✽ ✴ ✼✵

✭✈❛r✐❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❛♥❞ ❜❛❧❛♥❝❡❞✱ t❤❡♥ ✺ ♠❛① ♠✐♥ ❇♦r❡❧ ✵ ✶ ✶ ♠✐♥ ✇❤❡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✳ ✷ lwc ✭❝♦♥t✐♥✉✐t②✮ H n → µ t❤❡♥ L n ⇒ L ✸ � ✭♦♣t✐♠❛❧✐t②✮ Θ ✐s ❜❛❧❛♥❝❡❞ ✐✛ ✐t ♠✐♥✐♠✐③❡s f ( ∂ Θ) d µ ❢♦r str✐❝t❧② ❝♦♥✈❡① ✹ f : [ ✵ , ∞ ) → 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✳ ✷ lwc ✭❝♦♥t✐♥✉✐t②✮ H n → µ t❤❡♥ L n ⇒ L ✸ � ✭♦♣t✐♠❛❧✐t②✮ Θ ✐s ❜❛❧❛♥❝❡❞ ✐✛ ✐t ♠✐♥✐♠✐③❡s f ( ∂ Θ) d µ ❢♦r str✐❝t❧② ❝♦♥✈❡① ✹ f : [ ✵ , ∞ ) → R ✳ ✭✈❛r✐❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✮ t ∈ R ❛♥❞ Θ ❜❛❧❛♥❝❡❞✱ t❤❡♥ ✺ � � � ˜ ( ∂ Θ − t ) + d µ = ♠❛① f ♠✐♥ d � µ − t fd µ ❇♦r❡❧ f ∈H ∗ → [ ✵ , ✶ ] ✇❤❡r❡ ˜ ✶ f ♠✐♥ ( H , e , i ) = | e | ♠✐♥ j ∈ e f ( H , j ) ✳ ❍❛❥❡❦ ❈❊ ❱❈ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✽ ✴ ✼✵

✶ ✿ t❤❡ ❛♠♦✉♥t ♦❢ ❡①tr❛ ❧♦❛❞ s♦ t❤❛t t❤❡ t♦t❛❧ ❧♦❛❞ ❜❡❝♦♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ρ T , i ( x ) = t♦t❛❧ ❧♦❛❞ ❛t i ✇✐t❤ ❜❛s❡❧♦❛❞ x i ✳ ✳ ✳ ✳ ✳ ✳ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✾ ✴ ✼✵

✶ ✿ t❤❡ ❛♠♦✉♥t ♦❢ ❡①tr❛ ❧♦❛❞ s♦ t❤❛t t❤❡ t♦t❛❧ ❧♦❛❞ ❜❡❝♦♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ρ T , i ( x ) = t♦t❛❧ ❧♦❛❞ ❛t i ✇✐t❤ ❜❛s❡❧♦❛❞ x ↓ x i ✳ ✳ ✳ ✳ ✳ ✳ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✾ ✴ ✼✵

✶ ✿ t❤❡ ❛♠♦✉♥t ♦❢ ❡①tr❛ ❧♦❛❞ s♦ t❤❛t t❤❡ t♦t❛❧ ❧♦❛❞ ❜❡❝♦♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ρ T , i ( x ) = t♦t❛❧ ❧♦❛❞ ❛t i ✇✐t❤ ❜❛s❡❧♦❛❞ x ↓ x i ✳ ✳ ✳ ✳ ✳ ✳ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✾ ✴ ✼✵

✶ ✿ t❤❡ ❛♠♦✉♥t ♦❢ ❡①tr❛ ❧♦❛❞ s♦ t❤❛t t❤❡ t♦t❛❧ ❧♦❛❞ ❜❡❝♦♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ρ T , i ( x ) = t♦t❛❧ ❧♦❛❞ ❛t i ✇✐t❤ ❜❛s❡❧♦❛❞ x ↓ x ρ ( x ) i x ✳ ✳ ✳ ✳ ✳ ✳ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✹✾ ✴ ✼✵

❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ρ T , i ( x ) = t♦t❛❧ ❧♦❛❞ ❛t i ✇✐t❤ ❜❛s❡❧♦❛❞ x ↓ x ρ ( x ) i x ✳ ✳ ✳ ✳ ✳ ✳ ↓ ? t i 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 e ✶ e ✷ j ✶ j ✷ j ✸ j ✹ T e ✶ , j ✶ T e ✶ , j ✷ T e ✷ , j ✸ T e ✷ , j ✹ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ρ − ✶ T , i ( t ) =? i e ✶ e ✷ j ✶ j ✷ j ✸ j ✹ T e ✶ , j ✶ T e ✶ , j ✷ T e ✷ , j ✸ T e ✷ , j ✹ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ↓ ? ρ − ✶ T , i ( t ) =? t i e ✶ e ✷ j ✶ j ✷ j ✸ j ✹ T e ✶ , j ✶ T e ✶ , j ✷ T e ✷ , j ✸ T e ✷ , j ✹ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ↓ ? ρ − ✶ T , i ( t ) =? t i e ✶ e ✷ t t t t j ✶ j ✷ j ✸ j ✹ T e ✶ , j ✶ T e ✶ , j ✷ T e ✷ , j ✸ T e ✷ , j ✹ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ↓ ? ρ − ✶ T , i ( t ) =? t i e ✶ e ✷ θ ( e ✶ , j ✶ ) = ρ − ✶ T e ✶ , j ✶ ( t ) t t t t j ✶ j ✷ j ✸ j ✹ T e ✶ , j ✶ T e ✶ , j ✷ T e ✷ , j ✸ T e ✷ , j ✹ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ↓ ? ρ − ✶ T , i ( t ) =? t i e ✶ e ✷ θ ( e ✶ , j ✶ ) = ρ − ✶ T e ✶ , j ✶ ( t ) t t t t j ✶ j ✷ j ✸ j ✹ T e ✶ , j ✶ T e ✶ , j ✷ T e ✷ , j ✸ T e ✷ , j ✹ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ↓ ? ρ − ✶ T , i ( t ) =? t i e ✶ e ✷ θ ( e ✶ , j ✶ ) = ρ − ✶ T e ✶ , j ✶ ( t ) t t t t j ✶ j ✷ j ✸ j ✹ T e ✶ , j ✶ T e ✶ , j ✷ T e ✷ , j ✸ T e ✷ , j ✹ j � = i ρ − ✶ θ ( e ✶ , i ) = ✶ − � T e ✶ , j ( t ) j ∈ e ✶ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

✶ ✶ ✶ ✶ ✵ ❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ↓ ? ρ − ✶ T , i ( t ) =? t i e ✶ e ✷ θ ( e ✶ , j ✶ ) = ρ − ✶ T e ✶ , j ✶ ( t ) t t t t j ✶ j ✷ j ✸ j ✹ T e ✶ , j ✶ T e ✶ , j ✷ T e ✷ , j ✸ T e ✷ , j ✹ j � = i ρ − ✶ θ ( e ✶ , i ) = ✶ − � T e ✶ , j ( t ) j ∈ e ✶ � � ρ − ✶ � � ρ − ✶ T , i ( t ) = t − ✶ − T e , j ( t ) e ∋ i j ∈ e j � = i ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

❘❡❝✉rs✐♦♥ ♦❢ ❘❡s♣♦♥s❡ ❋✉♥❝t✐♦♥ ↓ ? ρ − ✶ T , i ( t ) =? t i e ✶ e ✷ θ ( e ✶ , j ✶ ) = ρ − ✶ T e ✶ , j ✶ ( t ) t t t t j ✶ j ✷ j ✸ j ✹ T e ✶ , j ✶ T e ✶ , j ✷ T e ✷ , j ✸ T e ✷ , j ✹ j � = i ρ − ✶ θ ( e ✶ , i ) = ✶ − � T e ✶ , j ( t ) j ∈ e ✶ � � ρ − ✶ � � ρ − ✶ T , i ( t ) = t − ✶ − T e , j ( t ) e ∋ i j ∈ e j � = i � ✶ � ρ − ✶ � � ρ − ✶ T e , j ( t ) + T , i ( t ) = t − ✶ − e ∋ i j ∈ e ✵ j � = i ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✵ ✴ ✼✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❞✐str✐❜✉t✐♦♥ ♦♥ ♥♦♥✕♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ✶ ✶ ❯●❲❚ ✐s ✉♥✐♠♦❞✉❧❛r ❯♥✐♠♦❞✉❧❛r ●❛❧t♦♥ ❲❛ts♦♥ ❍②♣❡rtr❡❡s ❆❧❧ t❤❡ ❤②♣❡r❡❞❣❡s ❤❛✈❡ s✐③❡ c ✭s❛② ✸✮ ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✶ ✴ ✼✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶ ❯●❲❚ ✐s ✉♥✐♠♦❞✉❧❛r ❯♥✐♠♦❞✉❧❛r ●❛❧t♦♥ ❲❛ts♦♥ ❍②♣❡rtr❡❡s ❆❧❧ t❤❡ ❤②♣❡r❡❞❣❡s ❤❛✈❡ s✐③❡ c ✭s❛② ✸✮ ❞✐str✐❜✉t✐♦♥ P ♦♥ ♥♦♥✕♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✶ ✴ ✼✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶ ❯●❲❚ ✐s ✉♥✐♠♦❞✉❧❛r ❯♥✐♠♦❞✉❧❛r ●❛❧t♦♥ ❲❛ts♦♥ ❍②♣❡rtr❡❡s ❆❧❧ t❤❡ ❤②♣❡r❡❞❣❡s ❤❛✈❡ s✐③❡ c ✭s❛② ✸✮ ❞✐str✐❜✉t✐♦♥ P ♦♥ ♥♦♥✕♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛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 ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛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 k = ( k + ✶ ) P k + ✶ ˆ E [ P ] P ˆ P ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛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 k = ( k + ✶ ) P k + ✶ ˆ E [ P ] P ˆ P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❱❡♥❦❛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 k = ( k + ✶ ) P k + ✶ ˆ E [ P ] P ˆ P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❯●❲❚ c ( P ) ∈ P ( H ∗ ) ✐s ✉♥✐♠♦❞✉❧❛r ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✶ ✴ ✼✵

♦♣t✐♠❛❧ ✶ ✶ ✵ ✵ ✶ ❯●❲❚ ✶ ✶ ✶ ✵ ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ✶ ✶ ▼❡❛♥ ❊①❝❡ss ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❯●❲❚ c ( P ) � � � ˜ ( ∂ Θ − t ) + d µ = s✉♣ f ♠✐♥ d � µ − t fd µ f : H ∗ → [ ✵ , ✶ ] ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✷ ✴ ✼✵

✶ ✵ ✵ ✶ ❯●❲❚ ✶ ✶ ✶ ✵ ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ✶ ✶ ▼❡❛♥ ❊①❝❡ss ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❯●❲❚ c ( P ) � � � ˜ ( ∂ Θ − t ) + d µ = s✉♣ f ♠✐♥ d � µ − t fd µ f : H ∗ → [ ✵ , ✶ ] ♦♣t✐♠❛❧ f = ✶ [ ∂ Θ > t ] ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✷ ✴ ✼✵

❯●❲❚ ✶ ✶ ✶ ✵ ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ✶ ✶ ▼❡❛♥ ❊①❝❡ss ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❯●❲❚ c ( P ) � � � ˜ ( ∂ Θ − t ) + d µ = s✉♣ f ♠✐♥ d � µ − t fd µ f : H ∗ → [ ✵ , ✶ ] ♦♣t✐♠❛❧ f = ✶ [ ∂ Θ > t ] ρ ( x ) ∂θ t ∂ Θ > t ⇔ ρ ( ✵ ) > t ⇔ ρ − ✶ ( t ) < ✵ x ρ − ✶ ( t ) ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✷ ✴ ✼✵

✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ✶ ✶ ▼❡❛♥ ❊①❝❡ss ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❯●❲❚ c ( P ) � � � ˜ ( ∂ Θ − t ) + d µ = s✉♣ f ♠✐♥ d � µ − t fd µ f : H ∗ → [ ✵ , ✶ ] ♦♣t✐♠❛❧ f = ✶ [ ∂ Θ > t ] ρ ( x ) ∂θ t ∂ Θ > t ⇔ ρ ( ✵ ) > t ⇔ ρ − ✶ ( t ) < ✵ x ρ − ✶ ( t ) ❯●❲❚ c ( P ) � � � ✶ t − � N ✶ − X + i , ✶ − · · · − X + � fd µ = P � ✵ < ✵ i = ✶ i , c − ✶ ✶ N X ✶ , ✶ X N , c − ✶ X ✶ , c − ✶ . . . . . . ❱❡♥❦❛t ❆♥❛♥t❤❛r❛♠ ▲❛r❣❡ ♥❡t✇♦r❦s ▼❛r❝❤ ✶✵✱ ✷✵✶✼ ✺✷ ✴ ✼✵