Asymptotic normality of the Parzen-Rosenblatt density estimator for strongly mixing random fields

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https://hal.archives-ouvertes.fr/hal-00530145

Preprint submitted on 27 Oct 2010

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Asymptotic normality of the Parzen-Rosenblatt density estimator for strongly mixing random fields

Mohamed El Machkouri

To cite this version:

Mohamed El Machkouri. Asymptotic normality of the Parzen-Rosenblatt density estimator for

strongly mixing random fields. 2010. �hal-00530145�

❆s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t② ♦❢ t❤❡ P❛r③❡♥✲❘♦s❡♥❜❧❛tt

❞❡♥s✐t② ❡st✐♠❛t♦r ❢♦r str♦♥❣❧② ♠✐①✐♥❣ r❛♥❞♦♠ ✜❡❧❞s

▼♦❤❛♠❡❞ ❊▲ ▼❆❈❍❑❖❯❘■✱

▲❛❜♦r❛t♦✐r❡ ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❘❛♣❤❛ë❧ ❙❛❧❡♠

❯▼❘ ❈◆❘❙ ✻✵✽✺✱ ❯♥✐✈❡rs✐té ❞❡ ❘♦✉❡♥

♠♦❤❛♠❡❞✳❡❧♠❛❝❤❦♦✉r✐❅✉♥✐✈✲r♦✉❡♥✳❢r

❖❝t♦❜❡r ✻✱ ✷✵✶✵

❆❜str❛❝t

❲❡ ♣r♦✈❡ t❤❡ ❛s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t② ♦❢ t❤❡ ❦❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t♦r ✭✐♥tr♦❞✉❝❡❞

❜② ❘♦s❡♥❜❧❛tt ✭✶✾✺✻✮ ❛♥❞ P❛r③❡♥ ✭✶✾✻✷✮✮ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ st❛t✐♦♥❛r② str♦♥❣❧②

♠✐①✐♥❣ r❛♥❞♦♠ ✜❡❧❞s✳ ❖✉r ❛♣♣r♦❛❝❤ ✐s ❜❛s❡❞ ♦♥ t❤❡ ▲✐♥❞❡❜❡r❣✬s ♠❡t❤♦❞ r❛t❤❡r t❤❛♥ ♦♥ ❇❡r♥st❡✐♥✬s s♠❛❧❧✲❜❧♦❝❦✲❧❛r❣❡✲❜❧♦❝❦ t❡❝❤♥✐q✉❡ ❛♥❞ ❝♦✉♣❧✐♥❣ ❛r❣✉♠❡♥ts

✇✐❞❡❧② ✉s❡❞ ✐♥ ♣r❡✈✐♦✉s ✇♦r❦s ♦♥ ♥♦♥♣❛r❛♠❡tr✐❝ ❡st✐♠❛t✐♦♥ ❢♦r s♣❛t✐❛❧ ♣r♦❝❡ss❡s✳

❖✉r ♠❡t❤♦❞ ❛❧❧♦✇s ✉s t♦ ❝♦♥s✐❞❡r ♦♥❧② ♠✐♥✐♠❛❧ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❜❛♥❞✇✐❞t❤ ♣❛✲

r❛♠❡t❡r ❛♥❞ ♣r♦✈✐❞❡s ❛ s✐♠♣❧❡ ❝r✐t❡r✐♦♥ ♦♥ t❤❡ str♦♥❣ ♠✐①✐♥❣ ❝♦❡✣❝✐❡♥ts ✇❤✐❝❤

❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❜❛♥❞✇✐t❤✳

❆▼❙ ❙✉❜❥❡❝t ❈❧❛ss✐✜❝❛t✐♦♥s ✭✷✵✵✵✮✿ ✻✷●✵✺✱ ✻✷●✵✼✱ ✻✵●✻✵✳

❑❡② ✇♦r❞s ❛♥❞ ♣❤r❛s❡s✿ ❈❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✱ ❦❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t♦r✱ str♦♥❣❧②

♠✐①✐♥❣ r❛♥❞♦♠ ✜❡❧❞s✱ s♣❛t✐❛❧ ♣r♦❝❡ss❡s✳

❙❤♦rt t✐t❧❡✿ ❑❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t✐♦♥ ❢♦r r❛♥❞♦♠ ✜❡❧❞s✳

✶ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ♠❛✐♥ r❡s✉❧t

❚❤❡ ❦❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t♦r ✐♥tr♦❞✉❝❡❞ ❜② ❘♦s❡♥❜❧❛tt ❬✷✵❪ ❛♥❞ P❛r③❡♥ ❬✶✽❪ ❤❛s r❡❝❡✐✈❡❞

❝♦♥s✐❞❡r❛❜❧❡ ❛tt❡♥t✐♦♥ ✐♥ ♥♦♥♣❛r❛♠❡tr✐❝ ❡st✐♠❛t✐♦♥ ♦❢ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t✐❡s ❢♦r t✐♠❡

s❡r✐❡s✳ ■❢ (X

i

)

i∈Z

✐s ❛ st❛t✐♦♥❛r② s❡q✉❡♥❝❡ ♦❢ r❡❛❧ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇✐t❤ ❛ ♠❛r❣✐♥❛❧

❞❡♥s✐t② f t❤❡♥ t❤❡ ❦❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t♦r ♦❢ f ✐s ❞❡✜♥❡❞ ❢♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n

❛♥❞ ❛♥② x ✐♥ R ❜②

f

_n

(x) = 1 nb

n

n

X

i=1

❑

x − X

i

b

n

✇❤❡r❡ ❑ ✐s ❛ ♣r♦❜❛❜✐❧✐t② ❦❡r♥❡❧ ❛♥❞ t❤❡ ❜❛♥❞✇✐❞t❤ b

n

✐s ❛ ♣❛r❛♠❡t❡r ✇❤✐❝❤ ❝♦♥✈❡r❣❡s s❧♦✇❧② t♦ ③❡r♦ s✉❝❤ t❤❛t nb

n

❣♦❡s t♦ ✐♥✜♥✐t② ✭t❤❡ ❜❛♥❞✇✐❞t❤ ❞❡t❡r♠✐♥❡s t❤❡ ❛♠♦✉♥t

♦❢ s♠♦♦t❤♥❡ss ♦❢ t❤❡ ❡st✐♠❛t♦r✮✳ ❋♦r s♠❛❧❧ b

n

✇❡ ❣❡t ❛ ✈❡r② r♦✉❣❤ ❡st✐♠❛t❡ ❛♥❞ ❢♦r

❧❛r❣❡ b

n

❛ s♠♦♦t❤ ❡st✐♠❛t❡✳ ❚❤❡ ❧✐t❡r❛t✉r❡ ❞❡❛❧✐♥❣ ✇✐t❤ t❤❡ ❛s②♠♣t♦t✐❝ ♣r♦♣❡rt✐❡s ♦❢ f

n

✇❤❡♥ t❤❡ ♦❜s❡r✈❛t✐♦♥s (X

i

)

i∈Z

❛r❡ ✐♥❞❡♣❡♥❞❡♥t ✐s ✈❡r② ❡①t❡♥s✐✈❡ ✭s❡❡ ❙✐❧✈❡r♠❛♥ ❬✷✶❪✮✳

■♥ ♣❛rt✐❝✉❧❛r✱ P❛r③❡♥ ❬✶✽❪ ♣r♦✈❡❞ t❤❛t ✇❤❡♥ (X

i

)

i∈Z

✐s ✐✳✐✳❞✳ ❛♥❞ t❤❡ ❜❛♥❞✇✐❞t❤ b

n

❣♦❡s t♦ ③❡r♦ s✉❝❤ t❤❛t nb

n

❣♦❡s t♦ ✐♥✜♥✐t② t❤❡♥ (nb

n

)

^1/2

(f

n

(x

0

) − E f

n

(x

0

)) ❝♦♥✈❡r❣❡s ✐♥

❞✐str✐❜✉t✐♦♥ t♦ t❤❡ ♥♦r♠❛❧ ❧❛✇ ✇✐t❤ ③❡r♦ ♠❡❛♥ ❛♥❞ ✈❛r✐❛♥❝❡ f (x

0

) R

R

❑

²

(t)dt✳ ❯♥❞❡r t❤❡ s❛♠❡ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❜❛♥❞✇✐❞t❤✱ t❤✐s r❡s✉❧t ✇❛s r❡❝❡♥t❧② ❡①t❡♥❞❡❞ ❜② ❲✉ ❬✷✹❪ ❢♦r

❝❛✉s❛❧ ❧✐♥❡❛r ♣r♦❝❡ss❡s ✇✐t❤ ✐✳✐✳❞✳ ✐♥♥♦✈❛t✐♦♥s ❛♥❞ ❜② ❉❡❞❡❝❦❡r ❛♥❞ ▼❡r❧❡✈è❞❡ ❬✼❪ ❢♦r str♦♥❣❧② ♠✐①✐♥❣ s❡q✉❡♥❝❡s✳ Pr❡✈✐♦✉s❧②✱ ❇♦sq✱ ▼❡r❧❡✈è❞❡ ❛♥❞ P❡❧✐❣r❛❞ ❬✸❪ ❡st❛❜❧✐s❤❡❞ ❛

❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠ ❢♦r t❤❡ ❦❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t♦r f

n

✇❤❡♥ t❤❡ s❡q✉❡♥❝❡ (X

i

)

i∈Z

✐s

❛ss✉♠❡❞ t♦ ❜❡ str♦♥❣❧② ♠✐①✐♥❣ ❜✉t t❤❡ ❜❛♥❞✇✐t❤ ♣❛r❛♠❡t❡r b

n

✐s ❛ss✉♠❡❞ t♦ s❛t✐s❢② b

n

≥ Cn

^−1/3

log n ✭❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t C✮ ✇❤✐❝❤ ✐s str♦♥❣❡r t❤❛♥ t❤❡ ❜❛♥❞✇✐t❤

♣❛r❛♠❡t❡r ❛ss✉♠♣t✐♦♥ ✐♥ ❬✶✽❪✱ ❬✼❪ ❛♥❞ ❬✷✹❪✳ ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❡st❛❜❧✐s❤

P❛r③❡♥✬s ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠ ✭s❡❡ ❚❤❡♦r❡♠ 1✮ ❢♦r r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇❤✐❝❤ s❤♦✇ s♣❛t✐❛❧

✐♥t❡r❛❝t✐♦♥ ✭r❛♥❞♦♠ ✜❡❧❞s✮✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s ♥♦t tr✐✈✐❛❧ s✐♥❝❡ Z

^d

❞♦❡s ♥♦t ❤❛✈❡ ❛ ♥❛t✉r❛❧

♦r❞❡r✐♥❣ ❢♦r d ≥ 2✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ♠❛rt✐♥❣❛❧❡✲❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞ ✭❲✉ ❬✷✹❪✮ ❢♦r t✐♠❡

s❡r✐❡s s❡❡♠s t♦ ❜❡ ❞✐✣❝✉❧t t♦ ❛♣♣❧② ❢♦r r❛♥❞♦♠ ✜❡❧❞s✳ ■♥ ♦r❞❡r t♦ ❡st❛❜❧✐s❤ ♦✉r ♠❛✐♥

r❡s✉❧t✱ ✇❡ ✉s❡ t❤❡ ▲✐♥❞❡❜❡r❣✬s ♠❡t❤♦❞ ✐♥tr♦❞✉❝❡❞ ✐♥ ✶✾✷✷ ❢♦r t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❝❡♥tr❛❧

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

s❡tt✐♥❣ ❜② ❉❡❞❡❝❦❡r ❬✻❪✳ ❖✉r ❛♣♣r♦❛❝❤ s❡❡♠s t♦ ❜❡ ❜❡tt❡r t❤❛♥ t❤❡ ❇❡r♥st❡✐♥✬s s♠❛❧❧✲

❜❧♦❝❦✲❧❛r❣❡✲❜❧♦❝❦ t❡❝❤♥✐q✉❡ ❛♥❞ ❝♦✉♣❧✐♥❣ ❛r❣✉♠❡♥ts ✇✐❞❡❧② ✉s❡❞ ✐♥ ♣r❡✈✐♦✉s ✇♦r❦s ♦♥

♥♦♥♣❛r❛♠❡tr✐❝ ❡st✐♠❛t✐♦♥ ❢♦r s♣❛t✐❛❧ ♣r♦❝❡ss❡s ✭s❡❡ ❬✹❪✱ ❬✺❪✱ ❬✶✷❪✱ ❬✷✷❪✮ s✐♥❝❡ ✇❡ ❛r❡ ❛❜❧❡

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

♦♥ t❤❡ str♦♥❣ ♠✐①✐♥❣ ❝♦❡✣❝✐❡♥ts ✇❤✐❝❤ ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ❜❛♥❞✇✐t❤✳ ❖✈❡r t❤❡ ❧❛st

❢❡✇ ②❡❛rs ♥♦♥♣❛r❛♠❡tr✐❝ ❡st✐♠❛t✐♦♥ ❢♦r r❛♥❞♦♠ ✜❡❧❞s ✭♦r s♣❛t✐❛❧ ♣r♦❝❡ss❡s✮ ✇❛s ❣✐✈❡♥

✐♥❝r❡❛s✐♥❣ ❛tt❡♥t✐♦♥ ✭s❡❡ ●✉②♦♥ ❬✶✶❪✮✳ ■♥ ❢❛❝t✱ s♣❛t✐❛❧ ❞❛t❛ ❛r✐s❡ ✐♥ ✈❛r✐♦✉s ❛r❡❛s ♦❢

r❡s❡❛r❝❤ ✐♥❝❧✉❞✐♥❣ ❡❝♦♥♦♠❡tr✐❝s✱ ✐♠❛❣❡ ❛♥❛❧②s✐s✱ ♠❡t❡♦r♦❧♦❣② ❛♥❞ ❣❡♦st❛t✐st✐❝s✳ ❙♦♠❡

❦❡② r❡❢❡r❡♥❝❡s ♦♥ ♥♦♥♣❛r❛♠❡tr✐❝ ❡st✐♠❛t✐♦♥ ❢♦r r❛♥❞♦♠ ✜❡❧❞s ❛r❡ ❇✐❛✉ ❬✶❪✱ ❈❛r❜♦♥ ❡t

❛❧✳ ❬✹❪✱ ❈❛r❜♦♥ ❡t ❛❧✳ ❬✺❪✱ ❍❛❧❧✐♥ ❡t ❛❧✳ ❬✶✷❪✱ ❬✶✸❪✱ ❚r❛♥ ❬✷✷❪✱ ❚r❛♥ ❛♥❞ ❨❛❦♦✇✐t③ ❬✷✸❪ ❛♥❞

❨❛♦ ❬✷✺❪ ✇❤♦ ❤❛✈❡ ✐♥✈❡st✐❣❛t❡❞ ♥♦♥♣❛r❛♠❡tr✐❝ ❞❡♥s✐t② ❡st✐♠❛t✐♦♥ ❢♦r r❛♥❞♦♠ ✜❡❧❞s ❛♥❞

✷

❇✐❛✉ ❛♥❞ ❈❛❞r❡ ❬✷❪✱ ❊❧ ▼❛❝❤❦♦✉r✐ ❬✾❪✱ ❊❧ ▼❛❝❤❦♦✉r✐ ❛♥❞ ❙t♦✐❝❛ ❬✶✵❪✱ ❍❛❧❧✐♥ ❡t ❛❧✳ ❬✶✹❪

❛♥❞ ▲✉ ❛♥❞ ❈❤❡♥ ❬✶✺❪✱ ❬✶✻❪ ✇❤♦ ❤❛✈❡ st✉❞✐❡❞ s♣❛t✐❛❧ ♣r❡❞✐❝t✐♦♥ ❛♥❞ s♣❛t✐❛❧ r❡❣r❡ss✐♦♥

❡st✐♠❛t✐♦♥✳

▲❡t d ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❛♥❞ ❧❡t (X

_i

)

_i∈Z^d

❜❡ ❛ ✜❡❧❞ ♦❢ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ r❡❛❧

r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇✐t❤ ❛ ♠❛r❣✐♥❛❧ ❞❡♥s✐t② f ✳ ●✐✈❡♥ t✇♦ σ ✲❛❧❣❡❜r❛s U ❛♥❞ V ♦❢ F ✱

❞✐✛❡r❡♥t ♠❡❛s✉r❡s ♦❢ t❤❡✐r ❞❡♣❡♥❞❡♥❝❡ ❤❛✈❡ ❜❡❡♥ ❝♦♥s✐❞❡r❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ❲❡ ❛r❡

✐♥t❡r❡st❡❞ ❜② ♦♥❡ ♦❢ t❤❡♠✳ ❚❤❡ α✲♠✐①✐♥❣ ❝♦❡✣❝✐❡♥t ❤❛s ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ ❜② ❘♦s❡♥❜❧❛tt

❬✷✵❪ ❞❡✜♥❡❞ ❜②

α( U , V ) = sup {| P (A ∩ B) − P (A) P (B ) | , A ∈ U , B ∈ V} .

■♥ t❤❡ s❡q✉❡❧✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ str♦♥❣ ♠✐①✐♥❣ ❝♦❡✣❝✐❡♥ts α

1,∞

(n) ❞❡✜♥❡❞ ❢♦r ❡❛❝❤ ♣♦s✐✲

t✐✈❡ ✐♥t❡❣❡r n ❜②

α

_1,∞

(n) = sup { α(σ(X

_k

), F

Γ

), k ∈ Z

^d

, Γ ⊂ Z

^d

, ρ(Γ, { k } ) ≥ n } ,

✇❤❡r❡ F

Γ

= σ(X

_i

; i ∈ Γ) ❛♥❞ t❤❡ ❞✐st❛♥❝❡ ρ ✐s ❞❡✜♥❡❞ ❢♦r ❛♥② s✉❜s❡ts Γ

₁

❛♥❞ Γ

₂

♦❢ Z

^d

❜② ρ(Γ

1

, Γ

2

) = min {| i − j | , i ∈ Γ

1

, j ∈ Γ

2

} ✇✐t❤ | i − j | = max

1≤s≤d

| i

s

− j

s

|

❢♦r ❛♥② i ❛♥❞ j ✐♥ Z

^d

✳ ❲❡ s❛② t❤❛t t❤❡ r❛♥❞♦♠ ✜❡❧❞ (X

i

)

_i∈Z^d

✐s str♦♥❣❧② ♠✐①✐♥❣ ✐❢

lim

n→+∞

α

1,∞

(n) = 0✳ ❚❤❡ ❝❧❛ss ♦❢ ♠✐①✐♥❣ r❛♥❞♦♠ ✜❡❧❞s ✐♥ t❤❡ ❛❜♦✈❡ s❡♥s❡ ✐s ✈❡r②

❧❛r❣❡ ✭♦♥❡ ❝❛♥ r❡❢❡r t♦ ●✉②♦♥ ❬✶✶❪ ♦r ❉♦✉❦❤❛♥ ❬✽❪ ❢♦r ❡①❛♠♣❧❡s✮ ❛♥❞ ✇❡ r❡❝❛❧❧ t❤❛t

❉❡❞❡❝❦❡r ❬✻❪ ♦❜t❛✐♥❡❞ ❛ ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠ ❢♦r t❤❡ st❛t✐♦♥❛r② r❛♥❞♦♠ ✜❡❧❞ (X

i

)

_i∈Z^d

♣r♦✈✐❞❡❞ t❤❛t X

0

❤❛s ③❡r♦ ♠❡❛♥ ❛♥❞ ✜♥✐t❡ ✈❛r✐❛♥❝❡ ❛♥❞

X

k∈Z^d

Z

α1,∞(|k|) 0

Q

²_X0

(u)du < + ∞

✇❤❡r❡ Q

X0

✐s t❤❡ q✉❛♥t✐❧❡ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ❢♦r ❛♥② u ✐♥ [0, 1] ❜② Q

X0

(u) = inf { t ≥ 0 ; P ( | X

0

| > t) ≤ u } .

❲❡ ❝♦♥s✐❞❡r t❤❡ ❞❡♥s✐t② ❡st✐♠❛t♦r ♦❢ f ❞❡✜♥❡❞ ❢♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n ❛♥❞ ❛♥② x ✐♥

R ❜②

f

n

(x) = 1 n

^d

b

n

X

i∈Λn

❑

x − X

i

b

n

✇❤❡r❡ b

_n

✐s t❤❡ ❜❛♥❞✇✐❞t❤ ♣❛r❛♠❡t❡r✱ Λ

_n

❞❡♥♦t❡s t❤❡ s❡t { 1, ..., n }

^d

❛♥❞ ❑ ✐s ❛ ♣r♦❜❛❜✐❧✲

✐t② ❦❡r♥❡❧✳ ❖✉r ❛✐♠ ✐s t♦ ♣r♦✈✐❞❡ ❛ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ str♦♥❣ ♠✐①✐♥❣ ❝♦❡✣❝✐❡♥ts α

1,∞

(n) ❢♦r (n

^d

b

n

)

^1/2

(f

n

(x

i

) − E f

n

(x

i

))

1≤i≤k

, (x

i

)

1≤i≤k

∈ R

^k

, k ∈ N

^∗

, t♦ ❝♦♥✈❡r❣❡ ✐♥ ❧❛✇

✸

t♦ ❛ ♠✉❧t✐✈❛r✐❛t❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✭❚❤❡♦r❡♠ 1✮ ✉♥❞❡r ♠✐♥✐♠❛❧ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡

❜❛♥❞✇✐❞t❤s ✭t❤❛t ✐s b

n

❣♦❡s t♦ ③❡r♦ ❛♥❞ n

^d

b

n

❣♦❡s t♦ ✐♥✜♥✐t②✮✳

❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss✉♠♣t✐♦♥s✿

✭❆✶✮ ❚❤❡ ♠❛r❣✐♥❛❧ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ♦❢ ❡❛❝❤ X

_k

✐s ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ✇✐t❤

❝♦♥t✐♥✉♦✉s ♣♦s✐t✐✈❡ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ f ✳

✭❆✷✮ ❚❤❡ ❥♦✐♥t ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ♦❢ ❡❛❝❤ (X

0

, X

k

) ✐s ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ✇✐t❤

❝♦♥t✐♥✉♦✉s ❥♦✐♥t ❞❡♥s✐t② f

_0,k

✳

✭❆✸✮ ❑ ✐s ❛ ♣r♦❜❛❜✐❧✐t② ❦❡r♥❡❧ ✇✐t❤ ❝♦♠♣❛❝t s✉♣♣♦rt ❛♥❞ R

R

❑

²

(u) du < ∞✳

✭❆✹✮ ❚❤❡ ❜❛♥❞✇✐❞t❤ b

n

❝♦♥✈❡r❣❡s t♦ ③❡r♦ ❛♥❞ n

^d

b

n

❣♦❡s t♦ ✐♥✜♥✐t②✳

❖✉r ♠❛✐♥ r❡s✉❧t ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳

❚❤❡♦r❡♠ ✶ ❆ss✉♠❡ t❤❛t ✭❆✶✮✱ ✭❆✷✮✱ ✭❆✸✮ ❛♥❞ ✭❆✹✮ ❤♦❧❞ ❛♥❞

+∞

X

m=1

m

^2d−1

α

1,∞

(m) < + ∞ . ✭✶✮

❚❤❡♥ ❢♦r ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r k ❛♥❞ ❛♥② ❞✐st✐♥❝t ♣♦✐♥ts x

1

, ..., x

k

✐♥ R ✱

(n

^d

b

_n

)

^1/2







f

n

(x

1

) − E f

n

(x

1

)

✳✳✳

f

n

(x

k

) − E f

n

(x

k

)







−−−−−→

L

n→+∞

N (0, V ) ✭✷✮

✇❤❡r❡ V ✐s ❛ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ✇✐t❤ ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts v

ii

= f (x

i

) R

R

❑

²

(u)du ✳

❘❡♠❛r❦ ✶✳ ❆ r❡♣❧❛❝❡♠❡♥t ♦❢ E f

_n

(x

_i

) ❜② f(x

_i

) ❢♦r ❛♥② 1 ≤ i ≤ k ✐♥ ✭ 2 ✮ ✐s ❛ ❝❧❛ss✐❝❛❧

♣r♦❜❧❡♠ ✐♥ ❞❡♥s✐t② ❡st✐♠❛t✐♦♥ t❤❡♦r②✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ f ✐s ❛ss✉♠❡❞ t♦ ❜❡ ▲✐♣s❝❤✐t③ ❛♥❞

✐❢ R

R

| u ||❑ (u) | du < ∞ t❤❡♥ | E f

n

(x

i

) − f(x

i

) | = O(b

n

) ❛♥❞ t❤✉s t❤❡ ❝❡♥t❡r✐♥❣ E f

n

(x

i

)

♠❛② ❜❡ ❝❤❛♥❣❡❞ t♦ f(x

i

) ✇✐t❤♦✉t ❛✛❡❝t✐♥❣ t❤❡ ❛❜♦✈❡ r❡s✉❧t ♣r♦✈✐❞❡❞ t❤❛t n

^d

b

³_n

❝♦♥✲

✈❡r❣❡s t♦ ③❡r♦✳

❘❡♠❛r❦ ✷✳ ❚❤❡♦r❡♠ 1 ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ❚❤❡♦r❡♠ ✸✳✶ ❜② ❇♦sq✱ ▼❡r❧❡✈è❞❡ ❛♥❞

P❡❧✐❣r❛❞ ❬✸❪✳ ■♥ ❢❛❝t✱ ✉s✐♥❣ ❛ ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤✱ t❤❡ ❛✉t❤♦rs ♦❜t❛✐♥❡❞ t❤❡ s❛♠❡ r❡s✉❧t

❢♦r d = 1 ✇✐t❤ ❛♥ ❛❞❞✐t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ❜❛♥❞✇✐t❤ ♣❛r❛♠❡t❡r ✭t❤❛t ✐s✱ t❤❡r❡ ❡①✐sts

❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t C s✉❝❤ t❤❛t b

_n

≥ C n

^−1/3

log n ✮✳ ❍♦✇❡✈❡r✱ t❤❡ ❝♦♥❞✐t✐♦♥ ✭✶✮ ✇✐t❤

d = 1 ✐s s❧✐❣❤t❧② ♠♦r❡ r❡str✐❝t✐✈❡ t❤❛♥ t❤❡ ❝♦♥❞✐t✐♦♥ P

m>n

α

_1,∞

(m) = o(n

⁻¹

) ♦❜t❛✐♥❡❞

❜② ❉❡❞❡❝❦❡r ❛♥❞ ▼❡r❧❡✈è❞❡ ✭❬✼❪✱ ❈♦r♦❧❧❛r② ✹✮✳ ❲❡ ❝♦♥❥❡❝t✉r❡ t❤❛t ❚❤❡♦r❡♠ ✶ st✐❧❧ ❤♦❧❞s

✉♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥ P

m>n

m

^d−1

α

1,∞

(m) = o(n

^−d

) ✳

✹

✷ Pr♦♦❢s

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✶✳ ❲✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ ✇❡ ❝♦♥s✐❞❡r ♦♥❧② t❤❡ ❝❛s❡ k = 2

❛♥❞ ✇❡ r❡❢❡r t♦ x

1

❛♥❞ x

2

❛s x ❛♥❞ y ✭x 6 = y✮✳ ▲❡t λ

1

❛♥❞ λ

2

❜❡ t✇♦ ❝♦♥st❛♥ts s✉❝❤

t❤❛t λ

²₁

+ λ

²₂

= 1 ❛♥❞ ❞❡♥♦t❡

S

n

= λ

1

(n

^d

b

n

)

^1/2

(f

n

(x) − E f

n

(x)) + λ

2

(n

^d

b

n

)

^1/2

(f

n

(y) − E f

n

(y)) = X

i∈Λn

∆

i

n

^d/2

✇❤❡r❡ ∆

i

= λ

1

Z

i

(x) + λ

2

Z

i

(y) ❛♥❞ ❢♦r ❛♥② z ✐♥ R ✱ Z

i

(z) = 1

√ b

n

❑

z − X

i

b

n

− E ❑

z − X

i

b

n

.

❲❡ ❝♦♥s✐❞❡r t❤❡ ♥♦t❛t✐♦♥s

η = (λ

²₁

f (x) + λ

²₂

f(y))σ

²

❛♥❞ σ

²

= Z

R

❑

²

(u)du. ✭✸✮

❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❡❝❤♥✐❝❛❧ r❡s✉❧t ✐s ♣♦st♣♦♥❡❞ t♦ t❤❡ ❛♥♥❡①✳

▲❡♠♠❛ ✶ E (∆

²₀

) ❝♦♥✈❡r❣❡s t♦ η ❛♥❞ E | ∆

0

∆

i

| = O(b

n

) ❢♦r ❛♥② i ✐♥ Z

^d

\{ 0 }✳

■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ ♦❢ S

n

t♦ √ ητ

0

✇❤❡r❡ τ

0

∼ N (0, 1)✱

✇❡ ❛r❡ ❣♦✐♥❣ t♦ ❢♦❧❧♦✇ t❤❡ ▲✐♥❞❡❜❡r❣✬s ♠❡t❤♦❞ ✉s❡❞ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠ ❢♦r st❛t✐♦♥❛r② r❛♥❞♦♠ ✜❡❧❞s ❜② ❉❡❞❡❝❦❡r ❬✻❪✳ ▲❡t ✉s ♥♦t❡ t❤❛t s❡✈❡r❛❧ ♣r❡✈✐✲

♦✉s ❛s②♠♣t♦t✐❝ r❡s✉❧ts ❢♦r ❦❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t❡s ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ s♣❛t✐❛❧ ♣r♦❝❡ss❡s

✇❡r❡ ❡st❛❜❧✐s❤❡❞ ✉s✐♥❣ t❤❡ s♦✲❝❛❧❧❡❞ ❇❡r♥st❡✐♥✬s s♠❛❧❧✲❜❧♦❝❦✲❧❛r❣❡✲❜❧♦❝❦ t❡❝❤♥✐q✉❡ ❛♥❞

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

✭s❡❡ ❢♦r ❡①❛♠♣❧❡ ❬✹❪✱ ❬✺❪✱ ❬✶✷❪✱ ❬✷✷❪✮✳ ❖✉r ❛♣♣r♦❛❝❤ s❡❡♠s t♦ ❜❡ ❜❡tt❡r s✐♥❝❡ ✇❡ ♦❜t❛✐♥

❛ ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠ ✇❤❡♥ t❤❡ ❜❛♥❞✇✐t❤ s❛t✐s✜❡s ♦♥❧② ❆ss✉♠♣t✐♦♥ ✭❆✹✮✳

▲❡t µ ❜❡ t❤❡ ❧❛✇ ♦❢ t❤❡ st❛t✐♦♥❛r② r❡❛❧ r❛♥❞♦♠ ✜❡❧❞ (X

k

)

_k∈Z^d

❛♥❞ ❝♦♥s✐❞❡r t❤❡ ♣r♦❥❡❝✲

t✐♦♥ π

0

❢r♦♠ R

^Zd

t♦ R ❞❡✜♥❡❞ ❜② π

0

(ω) = ω

0

❛♥❞ t❤❡ ❢❛♠✐❧② ♦❢ tr❛♥s❧❛t✐♦♥ ♦♣❡r❛t♦rs (T

^k

)

_k∈Z^d

❢r♦♠ R

^Zd

t♦ R

^Zd

❞❡✜♥❡❞ ❜② (T

^k

(ω))

i

= ω

i+k

❢♦r ❛♥② k ∈ Z

^d

❛♥❞ ❛♥② ω ✐♥ R

^Zd

✳

❉❡♥♦t❡ ❜② B t❤❡ ❇♦r❡❧ σ✲❛❧❣❡❜r❛ ♦❢ R ✳ ❚❤❡ r❛♥❞♦♠ ✜❡❧❞ (π

0

◦ T

^k

)

_k∈Z^d

❞❡✜♥❡❞ ♦♥ t❤❡

♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ( R

^Zd

, B

^Zd

, µ) ✐s st❛t✐♦♥❛r② ✇✐t❤ t❤❡ s❛♠❡ ❧❛✇ ❛s (X

_k

)

_k∈Z^d

✱ ❤❡♥❝❡✱ ✇✐t❤✲

♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ ♦♥❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t (Ω, F , P ) = ( R

^Zd

, B

^Zd

, µ) ❛♥❞ X

_k

= π

₀

◦ T

^k

✳

❖♥ t❤❡ ❧❛tt✐❝❡ Z

^d

✇❡ ❞❡✜♥❡ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r ❛s ❢♦❧❧♦✇s✿ ✐❢ i = (i

1

, ..., i

d

) ❛♥❞

j = (j

1

, ..., j

d

) ❛r❡ ❞✐st✐♥❝t ❡❧❡♠❡♥ts ♦❢ Z

^d

✱ t❤❡ ♥♦t❛t✐♦♥ i <

_❧❡①

j ♠❡❛♥s t❤❛t ❡✐t❤❡r

✺

i

1

< j

1

♦r ❢♦r s♦♠❡ p ✐♥ { 2, 3, ..., d }✱ i

p

< j

p

❛♥❞ i

q

= j

q

❢♦r 1 ≤ q < p✳ ▲❡t t❤❡ s❡ts { V

_i^M

; i ∈ Z

^d

, M ∈ N

^∗

} ❜❡ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿

V

_i¹

= { j ∈ Z

^d

; j <

_❧❡①

i } ,

❛♥❞ ❢♦r M ≥ 2

V

_i^M

= V

_i¹

∩ { j ∈ Z

^d

; | i − j | ≥ M } ✇❤❡r❡ | i − j | = max

1≤l≤d

| i

l

− j

l

| .

❋♦r ❛♥② s✉❜s❡t Γ ♦❢ Z

^d

❞❡✜♥❡ F

Γ

= σ(X

i

; i ∈ Γ) ❛♥❞ s❡t E

_M

(X

i

) = E (X

i

|F

V_i^M

), M ∈ N

^∗

.

▲❡t g ❜❡ ❛ ♦♥❡ t♦ ♦♥❡ ♠❛♣ ❢r♦♠ [1, M ] ∩ N

^∗

t♦ ❛ ✜♥✐t❡ s✉❜s❡t ♦❢ Z

^d

❛♥❞ (ξ

_i

)

_i∈Z^d

❛ r❡❛❧

r❛♥❞♦♠ ✜❡❧❞✳ ❋♦r ❛❧❧ ✐♥t❡❣❡rs k ✐♥ [1, M ] ✱ ✇❡ ❞❡♥♦t❡

S

g(k)

(ξ) =

k

X

i=1

ξ

g(i)

❛♥❞ S

_g(k)^c

(ξ) =

M

X

i=k

ξ

g(i)

✇✐t❤ t❤❡ ❝♦♥✈❡♥t✐♦♥ S

g(0)

(ξ) = S

_g(M+1)^c

(ξ) = 0✳ ❚♦ ❞❡s❝r✐❜❡ t❤❡ s❡t Λ

n

= { 1, ..., n }

^d

✱ ✇❡

❞❡✜♥❡ t❤❡ ♦♥❡ t♦ ♦♥❡ ♠❛♣ g ❢r♦♠ [1, n

^d

] ∩ N

^∗

t♦ Λ

n

❜②✿ g ✐s t❤❡ ✉♥✐q✉❡ ❢✉♥❝t✐♦♥ s✉❝❤

t❤❛t g(k) <

_❧❡①

g(l) ❢♦r 1 ≤ k < l ≤ n

^d

✳ ❋r♦♠ ♥♦✇ ♦♥✱ ✇❡ ❝♦♥s✐❞❡r ❛ ✜❡❧❞ (τ

_i

)

_i∈Z^d

♦❢

✐✳✐✳❞✳ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✐♥❞❡♣❡♥❞❡♥t ♦❢ (X

_i

)

_i∈Z^d

s✉❝❤ t❤❛t τ

₀

❤❛s t❤❡ st❛♥❞❛r❞ ♥♦r♠❛❧

❧❛✇ N (0, 1) ✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ✜❡❧❞s Y ❛♥❞ γ ❞❡✜♥❡❞ ❢♦r ❛♥② i ✐♥ Z

^d

❜② Y

i

= ∆

i

n

^d/2

❛♥❞ γ

i

= τ

i

√ η n

^d/2

✇❤❡r❡ η ✐s ❞❡✜♥❡❞ ❜② ✭ 3 ✮✳

▲❡t h ❜❡ ❛♥② ❢✉♥❝t✐♦♥ ❢r♦♠ R t♦ R ✳ ❋♦r 0 ≤ k ≤ l ≤ n

^d

+ 1 ✱ ✇❡ ✐♥tr♦❞✉❝❡

h

k,l

(Y ) = h(S

g(k)

(Y ) + S

_g(l)^c

(γ)) ✳ ❲✐t❤ t❤❡ ❛❜♦✈❡ ❝♦♥✈❡♥t✐♦♥ ✇❡ ❤❛✈❡ t❤❛t h

_k,n^d₊₁

(Y ) = h(S

g(k)

(Y )) ❛♥❞ ❛❧s♦ h

0,l

(Y ) = h(S

_g(l)^c

(γ))✳ ■♥ t❤❡ s❡q✉❡❧✱ ✇❡ ✇✐❧❧ ♦❢t❡♥ ✇r✐t❡ h

k,l

✐♥st❡❛❞

♦❢ h

k,l

(Y )✳ ❲❡ ❞❡♥♦t❡ ❜② B

₁⁴

( R ) t❤❡ ✉♥✐t ❜❛❧❧ ♦❢ C

_b⁴

( R )✿ h ❜❡❧♦♥❣s t♦ B

₁⁴

( R ) ✐❢ ❛♥❞ ♦♥❧②

✐❢ ✐t ❜❡❧♦♥❣s t♦ C

⁴

( R ) ❛♥❞ s❛t✐s✜❡s max

0≤i≤4

k h

⁽ⁱ⁾

k

∞

≤ 1✳

■t s✉✣❝❡s t♦ ♣r♦✈❡ t❤❛t ❢♦r ❛❧❧ h ✐♥ B

₁⁴

( R )✱

E h S

_g(n^d₎

(Y )

−−−−−→

n→+∞

E (h (τ

₀

√ η)) .

✻

❲❡ ✉s❡ ▲✐♥❞❡❜❡r❣✬s ❞❡❝♦♠♣♦s✐t✐♦♥✿

E h S

_g(n^d₎

(Y )

− h (τ

0

√ η)

=

n^d

X

k=1

E (h

k,k+1

− h

k−1,k

) .

◆♦✇✱

h

k,k+1

− h

k−1,k

= h

k,k+1

− h

k−1,k+1

+ h

k−1,k+1

− h

k−1,k

.

❆♣♣❧②✐♥❣ ❚❛②❧♦r✬s ❢♦r♠✉❧❛ ✇❡ ❣❡t t❤❛t✿

h

k,k+1

− h

k−1,k+1

= Y

g(k)

h

^′_k−1,k+1

+ 1

2 Y

_g(k)²

h

^′′_k−1,k+1

+ R

k

❛♥❞

h

k−1,k+1

− h

k−1,k

= − γ

g(k)

h

^′_k−1,k+1

− 1

2 γ

_g(k)²

h

^′′_k−1,k+1

+ r

k

✇❤❡r❡ | R

k

| ≤ Y

_g(k)²

(1 ∧ | Y

g(k)

| ) ❛♥❞ | r

k

| ≤ γ

_g(k)²

(1 ∧ | γ

g(k)

| )✳ ❙✐♥❝❡ (Y, τ

i

)

i6=g(k)

✐s ✐♥❞❡♣❡♥✲

❞❡♥t ♦❢ τ

g(k)

✱ ✐t ❢♦❧❧♦✇s t❤❛t E

γ

_g(k)

h

^′_k−1,k+1

= 0 ❛♥❞ E

γ

_g(k)²

h

^′′_k−1,k+1

= E η

n

^d

h

^′′_k−1,k+1

❍❡♥❝❡✱ ✇❡ ♦❜t❛✐♥

E h(S

_g(n^d₎

(Y )) − h (τ

₀

√ η)

=

n^d

X

k=1

E (Y

_g(k)

h

^′_k−1,k+1

)

+

n^d

X

k=1

E

Y

_g(k)²

− η n

^d

h

^′′_k−1,k+1

2

!

+

n^d

X

k=1

E (R

k

+ r

k

) .

▲❡t 1 ≤ k ≤ n

^d

❜❡ ✜①❡❞✳ ◆♦t✐♥❣ t❤❛t ∆

0

✐s ❜♦✉♥❞❡❞ ❜② 4 k❑k

∞

/ √

b

n

❛♥❞ ❛♣♣❧②✐♥❣

▲❡♠♠❛ 1 ✱ ✇❡ ❞❡r✐✈❡

E | R

_k

| ≤ E | ∆

₀

|

³

n

^3d/2

= O

1 (n

^3d

b

n

)

^1/2

❛♥❞

E | r

k

| ≤ E | γ

0

|

³

n

^3d/2

≤ η

^3/2

E | τ

0

|

³

n

^3d/2

= O

1 n

^3d/2

.

❈♦♥s❡q✉❡♥t❧②✱ ✇❡ ♦❜t❛✐♥

n^d

X

k=1

E ( | R

k

| + | r

k

| ) = O

1

(n

^d

b

_n

)

^1/2

+ 1 n

^d/2

= o(1).

✼

◆♦✇✱ ✐t ✐s s✉✣❝✐❡♥t t♦ s❤♦✇

n→+∞

lim

n^d

X

k=1

E (Y

g(k)

h

^′_k−1,k+1

) + E

Y

_g(k)²

− η n

^d

h

^′′_k−1,k+1

2

!!

= 0. ✭✹✮

❋✐rst✱ ✇❡ ❢♦❝✉s ♦♥ P

n^d

k=1

E Y

_g(k)

h

^′_k−1,k+1

✳ ❋♦r ❛❧❧ M ✐♥ N

^∗

❛♥❞ ❛❧❧ ✐♥t❡❣❡r k ✐♥ [1, n

^d

] ✱

✇❡ ❞❡✜♥❡

E

_k^M

= g([1, k] ∩ N

^∗

) ∩ V

_g(k)^M

❛♥❞ S

_g(k)^M

(Y ) = X

i∈E_k^M

Y

i

.

❋♦r ❛♥② ❢✉♥❝t✐♦♥ Ψ ❢r♦♠ R t♦ R ✱ ✇❡ ❞❡✜♥❡ Ψ

^M_k−1,l

= Ψ(S

_g(k)^M

(Y ) + S

_g(l)^c

(γ)) ✭✇❡ ❛r❡

❣♦✐♥❣ t♦ ❛♣♣❧② t❤✐s ♥♦t❛t✐♦♥ t♦ t❤❡ s✉❝❝❡ss✐✈❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ h✮✳

❋♦r ❛♥② ✐♥t❡❣❡r n✱ ✇❡ ❞❡✜♥❡

m

_n

= max



 

 



 

 



b

n^−12d

,





















 1 b

²_n

X

|i|>

» b

−12d n

–

| i |

^d

α

_1,∞

( | i | )











1 2d











 + 1



 

 



 

 



✇❤❡r❡ [ . ] ❞❡♥♦t❡s t❤❡ ✐♥t❡❣❡r ♣❛rt ❢✉♥❝t✐♦♥✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ t❡❝❤♥✐❝❛❧ ❧❡♠♠❛ ✐s t❤❡

s♣❛t✐❛❧ ✈❡rs✐♦♥ ♦❢ ❛ r❡s✉❧t ❜② ❇♦sq✱ ▼❡r❧❡✈è❞❡ ❛♥❞ P❡❧✐❣r❛❞ ✭❬✸❪✱ ♣❛❣❡s ✽✽✲✽✾✮✳ ■♥ ♦r❞❡r t♦ ❜❡ s❡❧❢✲❝♦♥t❛✐♥❡❞✱ t❤❡ ♣r♦♦❢ ✐s ❞♦♥❡ ✐♥ t❤❡ ❛♣♣❡♥❞✐①✳

▲❡♠♠❛ ✷ ❯♥❞❡r ❆ss✉♠♣t✐♦♥ ✭❆✹✮ ❛♥❞ t❤❡ ♠✐①✐♥❣ ❝♦♥❞✐t✐♦♥ (1) ✱ ✇❡ ❤❛✈❡

m

^d_n

→ ∞ , m

^d_n

b

n

→ 0 ❛♥❞ 1 m

^d_n

b

n

X

|i|>mn

| i |

^d

α

1,∞

( | i | ) → 0. ✭✺✮

❖✉r ❛✐♠ ✐s t♦ s❤♦✇ t❤❛t

n→+∞

lim

n^d

X

k=1

E

Y

g(k)

h

^′_k−1,k+1

− Y

g(k)

S

g(k−1)

(Y ) − S

_g(k)^mn

(Y )

h

^′′_k−1,k+1

= 0.

❋✐rst✱ ✇❡ ✉s❡ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥

Y

g(k)

h

^′_k−1,k+1

= Y

g(k)

h

^′_k−1,k+1^mn

+ Y

g(k)

h

^′_k−1,k+1

− h

^′_k−1,k+1^mn

.

❲❡ ❝♦♥s✐❞❡r ❛ ♦♥❡ t♦ ♦♥❡ ♠❛♣ m ❢r♦♠ [1, | E

_k^mn

| ] ∩ N

^∗

t♦ E

_k^mn

❛♥❞ s✉❝❤ t❤❛t | m(i) − g(k) | ≤ | m(i − 1) − g(k) |✳ ❚❤✐s ❝❤♦✐❝❡ ♦❢ m ❡♥s✉r❡s t❤❛t S

m(i)

(Y ) ❛♥❞ S

m(i−1)

(Y ) ❛r❡

F

_V|m(i)−g(k)|

g(k)

✲♠❡❛s✉r❛❜❧❡✳ ❚❤❡ ❢❛❝t t❤❛t γ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ Y ✐♠♣❧② t❤❛t E

Y

_g(k)

h

^′

S

_g(k+1)^c

(γ )

= 0.

✽

❚❤❡r❡❢♦r❡

E

Y

g(k)

h

^′_k−1,k+1^mn

=

|E^mn_k |

X

i=1

E Y

g(k)

(θ

i

− θ

i−1

)

✭✻✮

✇❤❡r❡ θ

i

= h

^′

S

_m(i)

(Y ) + S

_g(k+1)^c

(γ)

✳

❙✐♥❝❡ S

m(i)

(Y ) ❛♥❞ S

m(i−1)

(Y ) ❛r❡ F

_V|m(i)−g(k)|

g(k)

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

❡①♣❡❝t❛t✐♦♥ ♦❢ Y

g(k)

✇✐t❤ r❡s♣❡❝t t♦ F

_V|m(i)−g(k)|

g(k)

✐♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ♦❢ ✭6✮✳ ❖♥ t❤❡

♦t❤❡r ❤❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ h

^′

✐s 1✲▲✐♣s❝❤✐t③✱ ❤❡♥❝❡

| θ

_i

− θ

_i−1

| ≤ | Y

_m(i)

| .

❈♦♥s❡q✉❡♥t❧②✱

E Y

g(k)

(θ

i

− θ

i−1

)

≤ E | Y

m(i)

E

|m(i)−g(k)|

Y

g(k)

|

❛♥❞

E

Y

g(k)

h

^′_k−1,k+1^mn

≤

|E^mn_k |

X

i=1

E | Y

m(i)

E

|m(i)−g(k)|

(Y

g(k)

) | .

❍❡♥❝❡✱

n^d

X

k=1

E

Y

g(k)

h

^′_k−1,k+1^mn

≤ 1 n

^d

n^d

X

k=1

|E^mn_k |

X

i=1

E | ∆

m(i)

E

|m(i)−g(k)|

(∆

g(k)

) |

≤ X

|j|≥mn

k ∆

j

E

_|j|

(∆

0

) k

1

.

❋♦r ❛♥② j ✐♥ Z

^d

✱ ✇❡ ❤❛✈❡

k ∆

j

E

_|j|

(∆

0

) k

¹

= ❈♦✈

| ∆

j

| I

_E

|j|(∆0)≥0

− I

_E

|j|(∆0)<0

, ∆

0

.

❙♦✱ ❛♣♣❧②✐♥❣ ❘✐♦✬s ❝♦✈❛r✐❛♥❝❡ ✐♥❡q✉❛❧✐t② ✭❝❢✳ ❬✶✾❪✱ ❚❤❡♦r❡♠ ✶✳✶✮✱ ✇❡ ♦❜t❛✐♥

k ∆

_j

E

_|j|

(∆

₀

) k

1

≤ 4

Z

α1,∞(|j|) 0

Q

²_∆0

(u)du

✇❤❡r❡ Q

∆0

✐s ❞❡✜♥❡❞ ❜② Q

∆0

(u) = inf { t ≥ 0 ; P ( | ∆

0

| > t) ≤ u } ❢♦r ❛♥② u ✐♥ [0, 1]✳

❙✐♥❝❡ ∆

0

✐s ❜♦✉♥❞❡❞ ❜② 4 k❑k

∞

/ √

b

n

✱ ✇❡ ❤❛✈❡

Q

_∆0

(u) ≤ 4 k❑k

∞

√ b

n

❛♥❞ k ∆

_j

E

_|j|

(∆

₀

) k

1

≤ 64 k❑k

²∞

b

n

α

_1,∞

( | j | ).

✾

❋✐♥❛❧❧②✱ ✇❡ ❞❡r✐✈❡

n^d

X

k=1

E

Y

_g(k)

h

^′_k−1,k+1^mn

≤ 64 k K k

²∞

b

n

X

|j|≥mn

α

_1,∞

( | j | )

≤ 64 k K k

²∞

m

^d_n

b

n

X

|j|≥mn

| j |

^d

α

1,∞

( | j | )

= o(1) ❜② ✭ 5 ✮ .

❆♣♣❧②✐♥❣ ❛❣❛✐♥ ❚❛②❧♦r✬s ❢♦r♠✉❧❛✱ ✐t r❡♠❛✐♥s t♦ ❝♦♥s✐❞❡r

Y

_g(k)

(h

^′_k−1,k+1

− h

^′_k−1,k+1^mn

) = Y

_g(k)

(S

_g(k−1)

(Y ) − S

_g(k)^mn

(Y ))h

^′′_k−1,k+1

+ R

^′_k

,

✇❤❡r❡ | R

^′_k

| ≤ 2 | Y

_g(k)

(S

_g(k−1)

(Y ) − S

_g(k)^mn

(Y ))(1 ∧ | S

_g(k−1)

(Y ) − S

_g(k)^mn

(Y ) | ) |✳ ❉❡♥♦t✐♥❣

W

n

= {− m

n

+ 1, ..., m

n

− 1 }

^d

❛♥❞ W

_n^∗

= W

n

\{ 0 }✱ ✐t ❢♦❧❧♦✇s t❤❛t

n^d

X

k=1

E | R

^′_k

| ≤ 2 E | ∆

0

| X

i∈Wn

| ∆

i

|

! 1 ∧ 1

n

^d/2

X

i∈Wn

| ∆

i

|

!!

= 2 E







∆

²₀

+ X

i∈W_n^∗

| ∆

0

∆

i

|



 1 ∧ 1 n

^d/2

X

i∈Wn

| ∆

i

|

! 



≤ 2 n

^d/2

X

i∈Wn

E (∆

²₀

| ∆

_i

| ) + 2 X

i∈W_n^∗

E | ∆

₀

∆

_i

|

≤ 8 k❑k

∞

(n

^d

b

n

)

^1/2

X

i∈Wn

E ( | ∆

0

∆

i

| ) + 2 X

i∈W_n^∗

E | ∆

0

∆

i

| s✐♥❝❡ ∆

0

≤ 4 k K k

∞

√ b

n

❛✳s✳

= 8 E (∆

²₀

) k❑k

∞

(n

^d

b

n

)

^1/2

+ 2

1 + 4 k❑k

∞

(n

^d

b

n

)

^1/2

X

i∈W_n^∗

E ( | ∆

0

∆

i

| )

= O

1

(n

^d

b

n

)

^1/2

+ m

^d_n

b

n

1 + 1

(n

^d

b

n

)

^1/2

✭❜② ▲❡♠♠❛ 1✮

= o(1) ❜② ✭5✮.

❙♦✱ ✇❡ ❤❛✈❡ s❤♦✇♥ t❤❛t

n→+∞

lim

n^d

X

k=1

E

Y

g(k)

h

^′_k−1,k+1

− Y

g(k)

(S

g(k−1)

− S

_g(k)^mn

)h

^′′_k−1,k+1

= 0.

■♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ✭4✮ ✐t r❡♠❛✐♥s t♦ ❝♦♥tr♦❧

F

₀

= E





n^d

X

k=1

h

^′′_k−1,k+1

Y

_g(k)²

2 + Y

_g(k)

S

_g(k−1)

(Y ) − S

_g(k)^mn

(Y )

− η 2n

^d

! 

 .

✶✵

❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s❡ts✿

Λ

^m_nⁿ

= { i ∈ Λ

n

; ρ( { i } , ∂Λ

n

) ≥ m

n

} ❛♥❞ I

_n^mn

= { 1 ≤ i ≤ n

^d

; g(i) ∈ Λ

^m_nⁿ

} ,

❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ Ψ ❢r♦♠ R

^Zd

t♦ R s✉❝❤ t❤❛t Ψ(∆) = ∆

²₀

+ X

i∈V₀¹∩Wn

2∆

0

∆

i

✇❤❡r❡ W

n

= {− m

n

+ 1, ..., m

n

− 1 }

^d

.

❋♦r 1 ≤ k ≤ n

^d

✱ ✇❡ s❡t D

_k⁽ⁿ⁾

= η − Ψ ◦ T

^g(k)

(∆)✳ ❇② ❞❡✜♥✐t✐♦♥ ♦❢ Ψ ❛♥❞ ♦❢ t❤❡ s❡t I

_n^mn

✱

✇❡ ❤❛✈❡ ❢♦r ❛♥② k ✐♥ I

_n^mn

Ψ ◦ T

^g(k)

(∆) = ∆

²_g(k)

+ 2∆

g(k)

(S

g(k−1)

(∆) − S

_g(k)^mn

(∆)).

❚❤❡r❡❢♦r❡ ❢♦r k ✐♥ I

_n^mn

D

_k⁽ⁿ⁾

n

^d

= η

n

^d

− Y

_g(k)²

− 2Y

g(k)

(S

g(k−1)

(Y ) − S

_g(k)^mn

(Y )).

❙✐♥❝❡ lim

_n→+∞

n

^−d

| I

_n^mn

| = 1 ✱ ✐t r❡♠❛✐♥s t♦ ❝♦♥s✐❞❡r

F

1

= E



 1 n

^d

n^d

X

k=1

h

^′′_k−1,k+1

D

⁽ⁿ⁾_k



 .

❆♣♣❧②✐♥❣ ▲❡♠♠❛ 1✱ ✇❡ ❤❛✈❡

F

1

≤ E



 1 n

^d

n^d

X

k=1

h

^′′_k−1,k+1

(∆

²_g(k)

− E (∆

²₀

))





+ | η − E (∆

²₀

) | + 2 X

j∈V₀¹∩Wn

E | ∆

0

∆

j

|

≤ E



 1 n

^d

n^d

X

k=1

h

^′′_k−1,k+1

(∆

²_g(k)

− E (∆

²₀

))





+ o(1) + O(m

^d_n

b

n

),

✐t s✉✣❝❡s t♦ ♣r♦✈❡ t❤❛t

F

₂

= E



 1 n

^d

n^d

X

k=1

h

^′′_k−1,k+1

(∆

²_g(k)

− E (∆

²₀

))





❣♦❡s t♦ ③❡r♦ ❛s n ❣♦❡s t♦ ✐♥✜♥✐t②✳ ▲❡t M > 0 ❜❡ ✜①❡❞✳ ❲❡ ❤❛✈❡ F

2

≤ F

₂^′

+ F

₂^′′

✇❤❡r❡

F

₂^′

= E



 1 n

^d

n^d

X

k=1

h

^′′_k−1,k+1

∆

²_g(k)

− E

_M

∆

²_g(k)





✶✶

❛♥❞

F

₂^′′

= E



 1 n

^d

n^d

X

k=1

h

^′′_k−1,k+1

E

_M

∆

²_g(k)

− E (∆

²₀

)





✇❤❡r❡ ✇❡ r❡❝❛❧❧ t❤❡ ♥♦t❛t✐♦♥ E

_M

∆

²_g(k)

= E

∆

²_g(k)

|F

V_g(k)^M

✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ✐s ❛

❙❡r✢✐♥❣ t②♣❡ ✐♥❡q✉❛❧✐t② ✇❤✐❝❤ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✶✼❪✳

▲❡♠♠❛ ✸ ▲❡t U ❛♥❞ V ❜❡ t✇♦ σ✲❛❧❣❡❜r❛s ❛♥❞ ❧❡t X ❜❡ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♠❡❛s✉r❛❜❧❡

✇✐t❤ r❡s♣❡❝t t♦ U ✳ ■❢ 1 ≤ p ≤ r ≤ ∞ t❤❡♥

k E (X |V ) − E (X) k

p

≤ 2(2

^1/p

+ 1) (α( U , V ))

^1p−1r

k X k

r

.

❆♣♣❧②✐♥❣ ▲❡♠♠❛ 3 ❛♥❞ ❦❡❡♣✐♥❣ ✐♥ ♠✐♥❞ t❤❛t ∆

0

✐s ❜♦✉♥❞❡❞ ❜② 4 k❑k

∞

/ √

b

n

✱ ✇❡ ❞❡r✐✈❡

F

₂^′′

≤ k E

_M

∆

²₀

− E (∆

²₀

) k

1

≤ 96 k❑k

²∞

b

n

α

1,∞

(M )

■♥ t❤❡ ♦t❤❡r ♣❛rt✱

F

₂^′

≤ 1 n

^d

n^d

X

k=1

J

_k¹

(M ) + J

_k²

(M)

✇❤❡r❡

J

_k¹

(M ) = E

h

^′′_k−1,k+1^M

◦ T

^−g(k)

∆

²₀

− E

_M

∆

²₀

= 0 s✐♥❝❡ h

^′′_k−1,k+1^M

◦ T

^−g(k)

✐s F

V₀^M

✲♠❡❛s✉r❛❜❧❡ ❛♥❞

J

_k²

(M ) = E

h

^′′_k−1,k+1

◦ T

^−g(k)

− h

^′′_k−1,k+1^M

◦ T

^−g(k)

∆

²₀

− E

_M

∆

²₀

≤ E







2 ∧ X

|i|<M

| ∆

i

| n

^d/2



 ∆

²₀





≤ 4 k ❑ k

^∞

E (∆

²₀

)

(n

^d

b

_n

)

^1/2

+ 4 k ❑ k

^∞

(n

^d

b

_n

)

^1/2

X

|i|<M i6=0

E | ∆

i

∆

0

| s✐♥❝❡ ∆

0

≤ 4 k √ K k

^∞

b

n

❛✳s✳

= O

1

(n

^d

b

n

)

^1/2

+ M

^d

√ b

n

n

^d/2

✭❜② ▲❡♠♠❛ 1 ✮

❙♦✱ ♣✉tt✐♥❣ M = b

−1

n2d−1

❛♥❞ ❦❡❡♣✐♥❣ ✐♥ ♠✐♥❞ t❤❛t P

m≥0

m

^2d−1

α

1,∞

(m) < + ∞✱ ✇❡

❞❡r✐✈❡

F

2

= O M

^2d−1

α

1,∞

(M ) + O



 1 + b

d−1

n2d−1

(n

^d

b

_n

)

^1/2



 = o(1).

❚❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ 1 ✐s ❝♦♠♣❧❡t❡✳

✶✷

✸ ❆♣♣❡♥❞✐①

Pr♦♦❢ ♦❢ ▲❡♠♠❛ 1 ✳ ❋♦r ❛♥② i ✐♥ Z

^d

❛♥❞ ❛♥② z ✐♥ R ✱ ✇❡ ♥♦t❡ ❑

i

(z) = ❑

z−Xi

bn

✳ ❙♦✱ ✐❢

s ❛♥❞ t ❜❡❧♦♥❣s t♦ R ✱ ✇❡ ❤❛✈❡

E (Z

0

(s)Z

0

(t)) = 1 b

n

E ( ❑

0

(s) ❑

0

(t)) − E ❑

0

(s) E ❑

0

(t)

❛♥❞

n→+∞

lim 1

b

_n

E (❑

0

(s)❑

0

(t)) = lim

n→+∞

Z

R

❑ (v) ❑

v + t − s b

_n

f (s − vb

n

)dv = δ

st

f(s) σ

²

✇❤❡r❡ δ

st

= 1 ✐❢ s = t ❛♥❞ δ

st

= 0 ✐❢ s 6 = t✳ ❲❡ ❤❛✈❡ ❛❧s♦

n→+∞

lim 1 b

n

E ❑

0

(s) E ❑

0

(t) = lim

n→+∞

b

_n

Z

R

❑ (v)f (s − vb

_n

)dv Z

R

❑ (w)f (t − wb

_n

)dw = 0.

❙♦✱ ✇❡ ♦❜t❛✐♥

E (∆

²₀

) = λ

²₁

E (Z

₀²

(x)) + λ

²₂

E (Z

₀²

(y)) + 2λ

₁

λ

₂

E (Z

₀

(x)Z

₀

(y)) −−−−−→

n→+∞

η.

▲❡t i 6 = 0 ❜❡ ✜①❡❞ ✐♥ Z

^d

✳ ❲❡ ❤❛✈❡

E | ∆

0

∆

i

| ≤ λ

²₁

E | Z

0

(x)Z

i

(x) | +λ

²₂

E | Z

0

(y)Z

i

(y) | +λ

1

λ

2

E | Z

0

(x)Z

i

(y) | +λ

1

λ

2

E | Z

0

(y)Z

i

(x) | .

✭✼✮

❋♦r ❛♥② s ❛♥❞ t ✐♥ R ✱

E | Z

0

(s)Z

i

(t) | ≤ 1 b

n

E

❑

0

(s)❑

i

(t) + 1

b

n

E ❑

0

(s)

E ❑

0

(t)

.

▼♦r❡♦✈❡r✱ ✉s✐♥❣ ❆ss✉♠♣t✐♦♥s ✭❆✷✮ ❛♥❞ ✭❆✸✮✱ ✇❡ ❤❛✈❡

1 b

n

E ❑

0

(s)

E ❑

0

(t)

= b

_n

Z

R

|❑ (u) | f (s − ub

_n

)du Z

R

|❑ (v) | f (t − vb

_n

)dv = O(b

_n

)

❛♥❞

1 b

n

E

❑

0

(s) ❑

i

(t) = b

_n

Z Z

R²

❑ (w

₁

) ❑ (w

₂

)

f

_0,i

(s − w

₁

b

_n

, t − w

₂

b

_n

)dw

₁

dw

₂

= O(b

_n

).

❙♦✱ ✇❡ ♦❜t❛✐♥ ❢♦r ❛♥② s ❛♥❞ t ✐♥ R

E | Z

₀

(s)Z

_i

(t) | = O(b

_n

). ✭✽✮

❚❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ 1 ✐s ❝♦♠♣❧❡t❡❞ ❜② ❝♦♠❜✐♥✐♥❣ ✭ 7 ✮ ❛♥❞ ✭ 8 ✮✳

✶✸

Pr♦♦❢ ♦❢ ▲❡♠♠❛ 2✳ ❲❡ ❢♦❧❧♦✇ t❤❡ ♣r♦♦❢ ❜② ❇♦sq✱ ▼❡r❧❡✈è❞❡ ❛♥❞ P❡❧✐❣r❛❞ ✭❬✸❪✱ ♣❛❣❡s

✽✽✲✽✾✮✳ ❋✐rst✱ m

^d_n

❣♦❡s t♦ ✐♥✜♥✐t② s✐♥❝❡ b

n

❣♦❡s t♦ ③❡r♦ ❛♥❞ m

n

≥ h b

⁻n^2d1

i ✳ ❋♦r ❛♥②

♣♦s✐t✐✈❡ ✐♥t❡❣❡r m ✱ ✇❡ ❝♦♥s✐❞❡r

ψ(m) = X

|i|>m

| i |

^d

α

1,∞

( | i | ).

❙✐♥❝❡ t❤❡ ♠✐①✐♥❣ ❝♦♥❞✐t✐♦♥ ✭ 1 ✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ P

k∈Z^d

| k |

^d

α

_1,∞

( | k | ) < ∞✱ ✇❡ ❦♥♦✇ t❤❛t ψ(m) ❝♦♥✈❡r❣❡s t♦ ③❡r♦ ❛s m ❣♦❡s t♦ ✐♥✜♥✐t②✳ ▼♦r❡♦✈❡r✱ ✇❡ ❤❛✈❡

m

^d_n

b

n

≤ max (

p b

n

, C

d

r ψ h

b

⁻n^2d1

i + 2

^d

b

n

!)

−−−−−→

n→+∞

0

✇❤❡r❡ C

d

✐s s♦♠❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❞✐♠❡♥s✐♦♥ d ✳ ❲❡ ❤❛✈❡ ❛❧s♦

m

^d_n

≥ 1 b

n

r ψ h

b

⁻n^2d1

i ≥ 1 b

n

p ψ (m

n

) s✐♥❝❡ h b

⁻n^2d1

i ≤ m

n

.

❋✐♥❛❧❧②✱ ✇❡ ♦❜t❛✐♥

1 m

^d_n

b

n

X

|i|>mn

| i |

^d

α

_1,∞

( | i | ) ≤ p

ψ(m

_n

) −−−−−→

n→+∞

0.

❚❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ 2 ✐s ❝♦♠♣❧❡t❡✳

❘❡❢❡r❡♥❝❡s

❬✶❪ ●✳ ❇✐❛✉✳ ❙♣❛t✐❛❧ ❦❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t✐♦♥✳ ▼❛t❤✳ ▼❡t❤♦❞s ❙t❛t✐st✳✱ ✶✷✭✹✮✿✸✼✶✕✸✾✵✱

✷✵✵✸✳

❬✷❪ ●✳ ❇✐❛✉ ❛♥❞ ❇✳ ❈❛❞r❡✳ ◆♦♥♣❛r❛♠❡tr✐❝ s♣❛t✐❛❧ ♣r❡❞✐❝t✐♦♥✳ ❙t❛t✳ ■♥❢❡r❡♥❝❡ ❙t♦❝❤✳

Pr♦❝❡ss✳✱ ✼✭✸✮✿✸✷✼✕✸✹✾✱ ✷✵✵✹✳

❬✸❪ ❉✳ ❇♦sq✱ ▼❡r❧❡✈è❞❡ ❋✳✱ ❛♥❞ ▼✳ P❡❧✐❣r❛❞✳ ❆s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t② ❢♦r ❞❡♥s✐t② ❦❡r♥❡❧

❡st✐♠❛t♦rs ✐♥ ❞✐s❝r❡t❡ ❛♥❞ ❝♦♥t✐♥✉♦✉s t✐♠❡✳ ❏✳ ▼✉❧t✐✈❛r✐❛t❡ ❆♥❛❧✳✱ ✻✽✿✼✽✕✾✺✱ ✶✾✾✾✳

❬✹❪ ▼✳ ❈❛r❜♦♥✱ ▼✳ ❍❛❧❧✐♥✱ ❛♥❞ ▲✳❚✳ ❚r❛♥✳ ❑❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t✐♦♥ ❢♦r r❛♥❞♦♠ ✜❡❧❞s✿

t❤❡ l

1

t❤❡♦r②✳ ❏♦✉r♥❛❧ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ ❙t❛t✐st✐❝s✱ ✻✿✶✺✼✕✶✼✵✱ ✶✾✾✻✳

❬✺❪ ▼✳ ❈❛r❜♦♥✱ ▲✳❚✳ ❚r❛♥✱ ❛♥❞ ❇✳ ❲✉✳ ❑❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t✐♦♥ ❢♦r r❛♥❞♦♠ ✜❡❧❞s✳

❙t❛t✐st✳ Pr♦❜❛❜✳ ▲❡tt✳✱ ✸✻✿✶✶✺✕✶✷✺✱ ✶✾✾✼✳

✶✹

❬✻❪ ❏✳ ❉❡❞❡❝❦❡r✳ ❆ ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠ ❢♦r st❛t✐♦♥❛r② r❛♥❞♦♠ ✜❡❧❞s✳ Pr♦❜❛❜✳ ❚❤❡♦r②

❘❡❧❛t✳ ❋✐❡❧❞s✱ ✶✶✵✿✸✾✼✕✹✷✻✱ ✶✾✾✽✳

❬✼❪ ❏✳ ❉❡❞❡❝❦❡r ❛♥❞ ❋✳ ▼❡r❧❡✈è❞❡✳ ◆❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❝♦♥❞✐✲

t✐♦♥❛❧ ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✳ ❆♥♥❛❧s ♦❢ Pr♦❜❛❜✐❧✐t②✱ ✸✵✭✸✮✿✶✵✹✹✕✶✵✽✶✱ ✷✵✵✷✳

❬✽❪ P✳ ❉♦✉❦❤❛♥✳ ▼✐①✐♥❣✿ ♣r♦♣❡rt✐❡s ❛♥❞ ❡①❛♠♣❧❡s✱ ✈♦❧✉♠❡ ✽✺✳ ▲❡❝t✉r❡ ◆♦t❡s ✐♥

❙t❛t✐st✐❝s✱ ❇❡r❧✐♥✱ ✶✾✾✹✳

❬✾❪ ▼✳ ❊❧ ▼❛❝❤❦♦✉r✐✳ ◆♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ❡st✐♠❛t✐♦♥ ❢♦r r❛♥❞♦♠ ✜❡❧❞s ✐♥ ❛

✜①❡❞✲❞❡s✐❣♥✳ ❙t❛t✳ ■♥❢❡r❡♥❝❡ ❙t♦❝❤✳ Pr♦❝❡ss✳✱ ✶✵✭✶✮✿✷✾✕✹✼✱ ✷✵✵✼✳

❬✶✵❪ ▼✳ ❊❧ ▼❛❝❤❦♦✉r✐ ❛♥❞ ❘✳ ❙✳ ❙t♦✐❝❛✳ ❆s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t② ♦❢ ❦❡r♥❡❧ ❡st✐♠❛t❡s ✐♥

❛ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ❢♦r r❛♥❞♦♠ ✜❡❧❞s✳ ❚♦ ❛♣♣❡❛r ✐♥ ❏♦✉r♥❛❧ ♦❢ ◆♦♥♣❛r❛♠❡tr✐❝

❙t❛t✐st✐❝s✱ ✷✵✶✵✳

❬✶✶❪ ❳✳ ●✉②♦♥✳ ❘❛♥❞♦♠ ✜❡❧❞s ♦♥ ❛ ◆❡t✇♦r❦✿ ▼♦❞❡❧✐♥❣✱ ❙t❛t✐st✐❝s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✳

❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✱ ✶✾✾✺✳

❬✶✷❪ ▼✳ ❍❛❧❧✐♥✱ ❩✳ ▲✉✱ ❛♥❞ ▲✳❚✳ ❚r❛♥✳ ❉❡♥s✐t② ❡st✐♠❛t✐♦♥ ❢♦r s♣❛t✐❛❧ ❧✐♥❡❛r ♣r♦❝❡ss❡s✳

❇❡r♥♦✉❧❧✐✱ ✼✿✻✺✼✕✻✻✽✱ ✷✵✵✶✳

❬✶✸❪ ▼✳ ❍❛❧❧✐♥✱ ❩✳ ▲✉✱ ❛♥❞ ▲✳❚✳ ❚r❛♥✳ ❉❡♥s✐t② ❡st✐♠❛t✐♦♥ ❢♦r s♣❛t✐❛❧ ♣r♦❝❡ss❡s✿ t❤❡ l

₁

t❤❡♦r②✳ ❏✳ ▼✉❧t✐✈❛r✐❛t❡ ❆♥❛❧✳✱ ✽✽✭✶✮✿✻✶✕✼✺✱ ✷✵✵✹✳

❬✶✹❪ ▼✳ ❍❛❧❧✐♥✱ ❩✳ ▲✉✱ ❛♥❞ ▲✳❚✳ ❚r❛♥✳ ▲♦❝❛❧ ❧✐♥❡❛r s♣❛t✐❛❧ r❡❣r❡ss✐♦♥✳ ❆♥♥✳ ❙t❛t✐st✳✱

✸✷✭✻✮✿✷✹✻✾✕✷✺✵✵✱ ✷✵✵✹✳

❬✶✺❪ ❩✳ ▲✉ ❛♥❞ ❳✳ ❈❤❡♥✳ ❙♣❛t✐❛❧ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ❡st✐♠❛t✐♦♥✿ ◆♦♥✲✐s♦tr♦♣✐❝

❝❛s❡✳ ❆❝t❛ ▼❛t❤❡♠❛t✐❝❛❡ ❆♣♣❧✐❝❛t❛❡ ❙✐♥✐❝❛✱ ❊♥❣❧✐s❤ s❡r✐❡s✱ ✶✽✿✻✹✶✕✻✺✻✱ ✷✵✵✷✳

❬✶✻❪ ❩✳ ▲✉ ❛♥❞ ❳✳ ❈❤❡♥✳ ❙♣❛t✐❛❧ ❦❡r♥❡❧ r❡❣r❡ss✐♦♥ ❡st✐♠❛t✐♦♥✿ ✇❡❛❦ ❝♦♥s✐st❡♥❝②✳ ❙t❛t✐st✳

Pr♦❜❛❜✳ ▲❡tt✳✱ ✻✽✿✶✷✺✕✶✸✻✱ ✷✵✵✹✳

❬✶✼❪ ❉✳ ▲✳ ▼❝▲❡✐s❤✳ ❆ ♠❛①✐♠❛❧ ✐♥❡q✉❛❧✐t② ❛♥❞ ❞❡♣❡♥❞❡♥t str♦♥❣ ❧❛✇s✳ ❆♥♥✳ Pr♦❜❛❜✳✱

✸✭✺✮✿✽✷✾✕✽✸✾✱ ✶✾✼✺✳

❬✶✽❪ ❊✳ P❛r③❡♥✳ ❖♥ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ ❛ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❛♥❞ t❤❡ ♠♦❞❡✳ ❆♥♥✳ ▼❛t❤✳

❙t❛t✐st✳✱ ✸✸✿✶✾✻✺✕✶✾✼✻✱ ✶✾✻✷✳

✶✺