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Maximal coupling of empirical copulas for discrete vectors

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HAL Id: hal-01016563

https://hal.archives-ouvertes.fr/hal-01016563

Preprint submitted on 30 Jun 2014

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Maximal coupling of empirical copulas for discrete vectors

Olivier P. Faugeras

To cite this version:

Olivier P. Faugeras. Maximal coupling of empirical copulas for discrete vectors. 2013. �hal-01016563�

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✜rst ✈❡rs✐♦♥✿ ❥✉♥❡ ✷✵✶✸✱ t❤✐s ✈❡rs✐♦♥✿ ❥✉♥❡ ✷✵✶✹✳

▼❛①✐♠❛❧ ❝♦✉♣❧✐♥❣ ♦❢ ❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛s ❢♦r ❞✐s❝r❡t❡ ✈❡❝t♦rs

❖❧✐✈✐❡r P✳ ❋❛✉❣❡r❛s

❚♦✉❧♦✉s❡ ❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s ✲ ❯♥✐✈❡rs✐té ❚♦✉❧♦✉s❡ ✶ ❈❛♣✐t♦❧❡ ✲ ●❘❊▼❆◗✱

▼❛♥✉❢❛❝t✉r❡ ❞❡s ❚❛❜❛❝s✱ ❇✉r❡❛✉ ▼❋✸✶✾✱ ✷✶ ❆❧❧é❡ ❞❡ ❇r✐❡♥♥❡✱ ✸✶✵✵✵ ❚♦✉❧♦✉s❡✱ ❋r❛♥❝❡

❆❜str❛❝t

❋♦r ❛ ✈❡❝t♦r X ✇✐t❤ ❛ ♣✉r❡❧② ❞✐s❝r❡t❡ ♠✉❧t✐✈❛r✐❛t❡ ❞✐str✐❜✉t✐♦♥✱ ✇❡ ❣✐✈❡ s✐♠♣❧❡ s❤♦rt

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

❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥s✱ ♦❜t❛✐♥❡❞ ❡✐t❤❡r ✈✐❛ ♣r♦❜❛❜✐❧✐st✐❝ ❝♦♥t✐♥✉❛t✐♦♥✱ ✐✳❡✳ ❦❡r♥❡❧

s♠♦♦t❤✐♥❣✱ ♦r ✈✐❛ t❤❡ ❞✐str✐❜✉t✐♦♥❛❧ tr❛♥s❢♦r♠✳ ❚❤❡s❡ r❡s✉❧ts ❣✐✈❡ ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ s♦♠❡ ❞❡❧✐❝❛t❡ ✐ss✉❡s r❡❧❛t❡❞ t♦ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ ❞✐s❝r❡t❡

❝❛s❡✳ ❚❤❡② ❛r❡ ♦❜t❛✐♥❡❞ ✉♥❞❡r t❤❡ ✈❡r② ✇❡❛❦ ❤②♣♦t❤❡s✐s ♦❢ ❡r❣♦❞✐❝✐t② ♦❢ t❤❡ s❛♠♣❧❡✱ ❛

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

▼♦r❡♦✈❡r✱ t❤❡② ❛❧❧♦✇ t♦ ❞❡r✐✈❡✱ ❛s ❛ s✐♠♣❧❡ ❝♦r♦❧❧❛r②✱ ❛❧♠♦st s✉r❡ ❝♦♥s✐st❡♥❝② r❡s✉❧ts

❢♦r s♦♠❡ r❡❝❡♥t ❡①t❡♥s✐♦♥s ♦❢ ❝♦♥❝♦r❞❛♥❝❡ ♠❡❛s✉r❡s ❛tt❛❝❤❡❞ t♦ ❞✐s❝r❡t❡ ✈❡❝t♦rs✳ ❚❤❡

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

✐♥❞❡♣❡♥❞❡♥t ✐♥t❡r❡st✳

❑❡②✇♦r❞s✿ ❉✐s❝r❡t❡ ✈❡❝t♦r✱ ▼❛①✐♠❛❧ ❝♦✉♣❧✐♥❣✱ ❛✳s✳ ❝♦♥str✉❝t✐♦♥s✱ ❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛✱

❡r❣♦❞✐❝✐t②✳

✷✵✶✵ ▼❙❈✿ ✻✵❊✵✺✱ ✻✷❍✵✺✱ ✻✷❍✶✷✱ ✻✷❊✷✵✳

✶✳ ■♥tr♦❞✉❝t✐♦♥

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

♦❢ ♠❛r❣✐♥❛❧ ❝❞❢s G = (G 1 , . . . , G d )✱ ♥❛♠❡❧② G i (x i ) = F (∞, . . . , ∞, x i , ∞, . . . , ∞)✳ ❲❡

❞❡♥♦t❡ ✈❡❝t♦rs ❜② ❜♦❧❞ ❧❡tt❡rs✱ ❛♥❞ ✐♥t❡r♣r❡t ♦♣❡r❛t✐♦♥s ❜❡t✇❡❡♥ ✈❡❝t♦rs ❝♦♠♣♦♥❡♥t✇✐s❡✳

▲❡t ||x|| 1 = P d

i=1 |x i | ❜❡ t❤❡ l 1 ♥♦r♠ ♦♥ R d ✱ ❛♥❞ ||.|| ∞ t❤❡ s✉♣r❡♠✉♠ ♥♦r♠ ♦♥ R ✱ s♦ t❤❛t

||||G|| ∞ || 1 =

d

X

i=1

sup

x

i

∈ R

|G i (x i )|.

❊♠❛✐❧ ❛❞❞r❡ss✿ ♦❧✐✈✐❡r✳❢❛✉❣❡r❛s❅ts❡✲❢r✳❡✉ ✭❖❧✐✈✐❡r P✳ ❋❛✉❣❡r❛s✮

Pr❡♣r✐♥t s✉❜♠✐tt❡❞ t♦ ❊❧s❡✈✐❡r ❏✉♥❡ ✷✼✱ ✷✵✶✹

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✶✳✶✳ ❈♦♣✉❧❛ ❢✉♥❝t✐♦♥s

❘❡❝❛❧❧ t❤❛t ❛ d✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥ C : [0, 1] d 7→ [0, 1] ✐s ❞❡✜♥❡❞ ❛♥❛❧②t✐❝❛❧❧②

❛s ❛ ❣r♦✉♥❞❡❞✱ d−✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✱ ✇✐t❤ ✉♥✐❢♦r♠ ♠❛r❣✐♥❛❧s ✇❤♦s❡ ❞♦♠❛✐♥ ✐s [0, 1] d

✭s❡❡ ◆❡❧s❡♥ ❬✶✷❪✮✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ✐t ❝❛♥ ❜❡ ❞❡✜♥❡❞ ♣r♦❜❛❜✐❧✐st✐❝❛❧❧② ❛s t❤❡ r❡str✐❝t✐♦♥ t♦

[0, 1] d ♦❢ t❤❡ ♠✉❧t✐✈❛r✐❛t❡ ❝❞❢ ♦❢ ❛ r❛♥❞♦♠ ✈❡❝t♦r U ✱ ❝❛❧❧❡❞ ❛ ❝♦♣✉❧❛ r❡♣r❡s❡♥t❡r✱ ✇❤♦s❡

♠❛r❣✐♥❛❧s ❛r❡ ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ♦♥ [0, 1] ✭s❡❡ ❘üs❝❤❡♥❞♦r❢ ❬✶✻✱ ✶✼❪✮✳ ❚❤❡✐r ✐♥t❡r❡st st❡♠s ❢r♦♠ ❙❦❧❛r✬s t❤❡♦r❡♠ ✭s❡❡ ❬✶✽✱ ✶✾❪✮✱ ✇❤✐❝❤ ❛ss❡rts t❤❛t✱ ❢♦r ❡✈❡r② r❛♥❞♦♠ ✈❡❝t♦r X ∼ F ✱ t❤❡r❡ ❡①✐sts ❛ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥ ❝♦♥♥❡❝t✐♥❣✱ ♦r ❛ss♦❝✐❛t❡❞ ✇✐t❤ X ✱ ✐♥ t❤❡ s❡♥s❡

t❤❛t✿

❚❤❡♦r❡♠ ✶✳✶✳ ❋♦r ❡✈❡r② ♠✉❧t✐✈❛r✐❛t❡ ❝❞❢ F✱ ✇✐t❤ ♠❛r❣✐♥❛❧ ❝❞❢s G ✱ t❤❡r❡ ❡①✐sts s♦♠❡

❝♦♣✉❧❛ ❢✉♥❝t✐♦♥ C s✉❝❤ t❤❛t

F(x) = C(G(x)), ∀x ∈ R d , ✭✶✮

✇❤❡r❡ G ( x ) = (G 1 (x 1 ), . . . , G d (x d ))✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ C ✐s ❛ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥ ❛♥❞ G ❛ s❡t ♦❢

♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s✱ t❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ F ❞❡✜♥❡❞ ❜② ✭✶✮ ✐s ❛ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥

❢✉♥❝t✐♦♥ ✇✐t❤ ♠❛r❣✐♥❛❧s G ✳

❲❤❡♥ G ✐s ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❝♦♣✉❧❛ C ❛ss♦❝✐❛t❡❞ ✇✐t❤ X ✐♥ r❡❧❛t✐♦♥ ✭✶✮ ✐s ✉♥✐q✉❡ ❛♥❞

❝❛♥ ❜❡ ❞❡✜♥❡❞ ❢r♦♠ F ❡✐t❤❡r ❛♥❛❧②t✐❝❛❧❧② ❜② C = F ◦ G −1 ✱ ✇❤❡r❡ G −1 = (G −1 1 , . . . , G −1 d )

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

❛t❡ Pr♦❜❛❜✐❧✐t② ■♥t❡❣r❛❧ ❚r❛♥s❢♦r♠✱ ♥❛♠❡❧② C(u) = P (G(X) ≤ u)✱ u ∈ [0, 1] d ✳ ❲❤❡♥❡✈❡r

❞✐s❝♦♥t✐♥✉✐t② ✐s ♣r❡s❡♥t✱ C ✐s ♥♦ ❧♦♥❣❡r ✉♥✐q✉❡✿ ✐♥ ♦t❤❡r ✇♦r❞s C✱ ❛s ❛ ❢✉♥❝t✐♦♥❛❧ ♣❛✲

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

❝♦♣✉❧❛ r❡♣r❡s❡♥t❡r U ❛ss♦❝✐❛t❡❞ ✇✐t❤ X ❝❛♥ ❜❡ ❜❛s❡❞ ♦♥✿

✐✮ t❤❡ ❞✲✈❛r✐❛t❡ ❞✐str✐❜✉t✐♦♥❛❧ tr❛♥s❢♦r♠ U = G(X, V) ✇❤❡r❡ G j (x j , λ) = P (X j <

x j ) + λP (X j = x j )✱ j = 1, . . . , d✱ λ ∈ [0, 1]✱ ❛♥❞ V ✐s ❛ ✈❡❝t♦r ♦❢ ✉♥✐❢♦r♠ [0, 1]

♠❛r❣✐♥❛❧s✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ X ✭s❡❡ ▼♦♦r❡ ❛♥❞ ❙♣r✉✐❧❧ ❬✶✶❪✱ ❘üs❝❤❡♥❞♦r❢ ❬✶✺✱ ✶✻✱ ✶✼❪✱

◆❡s❧❡❤♦✈❛ ❬✶✸❪✮❀

✐✐✮ ♣r♦❜❛❜✐❧✐st✐❝ ❝♦♥t✐♥✉❛t✐♦♥✱ ✐✳❡✳ ❜② t❛❦✐♥❣ t❤❡ ❧✐♠✐t ♦❢ U h = G ˆ h (X h ) ✐♥ ❞✐str✐❜✉t✐♦♥

❛❧♦♥❣ ❛ s✉❜s❡q✉❡♥❝❡✱ ✇❤❡r❡ G ˆ h ✐s t❤❡ ✈❡❝t♦r ♦❢ ♠❛r❣✐♥❛❧ ❝❞❢ ♦❢ t❤❡ ❝♦♥t✐♥✉❡❞

X h = X + h Z ✱ ✇❤❡r❡ Z ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ h ↓ 0 ✭s❡❡ ❋❛✉❣❡r❛s ❬✻❪✮✳

✶✳✷✳ ❊♠♣✐r✐❝❛❧ ❝♦♣✉❧❛s ❢♦r ❝♦♥t✐♥✉♦✉s ❞✐str✐❜✉t✐♦♥s

■❢ F ✐s ✉♥❦♥♦✇♥✱ ❜✉t ♦♥❡ ❤❛s ✐♥st❡❛❞ ❛ s❛♠♣❧❡ X 1 , X 2 , . . . ♦❢ ❝♦♣✐❡s ❞✐str✐❜✉t❡❞ ❛❝✲

❝♦r❞✐♥❣ t♦ F ♦♥ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω, A, P )✱ ♦♥❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❡❝❞❢ F n ✱

F n (x) = 1 n

n

X

i=1

1 X

i

≤ x ,

❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❡❝t♦r ♦❢ ♠❛r❣✐♥❛❧ ❡❝❞❢s G n . ❙❦❧❛r✬s t❤❡♦r❡♠ t❤❡r❡❢♦r❡ ❡♥t❛✐❧s t❤❛t t❤❡r❡ ❡①✐sts s♦♠❡ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥ C n ❛ss♦❝✐❛t❡❞ ✇✐t❤ F n ✳ ❆s t❤❡ ❡❝❞❢ ✐s ❞✐s❝r❡t❡✱ C n ✐s

♥♦ ❧♦♥❣❡r ✉♥✐q✉❡ ❛♥❞ ❝❛♥ ♥♦ ❧♦♥❣❡r ❜❡ ❞❡✜♥❡❞✱ ✐♥ ♣❛r❛❧❧❡❧ ✇✐t❤ t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✱ ❛s C n := F n ◦ G −1 n , ♥♦r ❛s C n ∗∗ (u) := P (G n (X n ) ≤ u), ✇✐t❤ X n ∼ F n ✱ ❝♦♥❞✐t✐♦♥❛❧❧② ♦♥

t❤❡ s❛♠♣❧❡✱ ❛♥❞ ✇❤❡r❡ P ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦❜❛❜✐❧✐t② ✭♠♦r❡ ♦♥ t❤✐s ❜❡❧♦✇✮✳ ■♥❞❡❡❞✱

C n ❛♥❞ C n ∗∗ ❞♦ ♥♦t ❤❛✈❡ ✉♥✐❢♦r♠ ♠❛r❣✐♥❛❧s ❛♥❞ ❤❡♥❝❡ ❛r❡ ♥♦t ❣❡♥✉✐♥❡ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥s

(4)

❛ss♦❝✐❛t❡❞ ✇✐t❤ F n ✳ C n ❛♥❞ C n ∗∗ ❛r❡ ✈❡rs✐♦♥s ♦❢ t❤❡ ✐♠♣r♦♣❡r❧② ❝❛❧❧❡❞ ❡♠♣✐r✐❝❛❧ ✏❝♦♣✉❧❛✑

❢✉♥❝t✐♦♥s✱ ✐♥tr♦❞✉❝❡❞ ❜② ❘üs❝❤❡♥❞♦r❢ ❬✶✹❪ ✉♥❞❡r t❤❡ ♥❛♠❡ ♦❢ ♠✉❧t✐✈❛r✐❛t❡ r❛♥❦ ♦r❞❡r

❢✉♥❝t✐♦♥ ❛♥❞ ❉❡❤❡✉✈❡❧s ❬✷✱ ✸❪ ✉♥❞❡r t❤❡ ♥❛♠❡ ♦❢ ❡♠♣✐r✐❝❛❧ ❞❡♣❡♥❞❡♥❝❡ ❢✉♥❝t✐♦♥✳

❲❤❡♥ F ✐s ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❞✐s❛❞✈❛♥t❛❣❡ ♦❢ ❡st✐♠❛t✐♥❣ C = F ◦ G −1 ❜② ❡st✐♠❛t♦rs

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

♣❛r❛♠❡t❡r t♦ ❜❡ ❡st✐♠❛t❡❞✱ ✐s ♠✐t✐❣❛t❡❞ ❜② t❤❡ ❢❛❝t t❤❛t t❤❡s❡ ❡st✐♠❛t♦rs ❝♦✐♥❝✐❞❡✱ ✇✐t❤

❛♥② ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F n ♦♥ t❤❡ ❣r✐❞ ♦❢ ♣♦✐♥ts u k = (k 1 /n, . . . , k d /n)

❢♦r k 1 , . . . , k d = 0, . . . , n❀ s❡❡ ❉❡❤❡✉✈❡❧s ❬✸❪✳ ▼♦r❡♦✈❡r✱ ❛♥② ✈❡rs✐♦♥ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣

❡♠♣✐r✐❝❛❧ ✏❝♦♣✉❧❛✑ ♣r♦❝❡ss ✇❡❛❦❧② ❝♦♥✈❡r❣❡s✱ s❡❡ ❡✳❣✳ ❋❡r♠❛♥✐❛♥ ❡t ❛❧✳ ❬✼❪✱ ♦r ❘üs❝❤❡♥❞♦r❢

❬✶✹❪✳ ❍❡♥❝❡✱ ✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✱ t❤❡ ❝❤♦✐❝❡ ♦❢ ✇❤✐❝❤ ✏❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛✑ ❢✉♥❝t✐♦♥ t♦

✉s❡ ✐s ♦❢t❡♥ ♦❢ ❧✐tt❧❡ r❡❧❡✈❛♥❝❡ ❢♦r st❛t✐st✐❝❛❧ ♣✉r♣♦s❡s✳

✶✳✸✳ ❊♠♣✐r✐❝❛❧ ❝♦♣✉❧❛s ❢♦r ❞✐s❝r❡t❡ ❞✐str✐❜✉t✐♦♥s

❚♦ t❤❡ ❝♦♥tr❛r②✱ ✇❤❡♥ F ❤❛s ❛ ❞✐s❝r❡t❡ ❝♦♠♣♦♥❡♥t✱ t❤❡ ✐♥❞❡t❡r♠✐♥❛❝② ♦❢ C ❛♥❞ ❤❡♥❝❡

♦❢ C n ✐s ♠♦r❡ ❛❝✉t❡✿ ❉❡❤❡✉✈❡❧s ❬✶❪ ❛♥❞ ▲✐♥❞♥❡r ❛♥❞ ❙③✐♠❛②❡r ❬✽❪ s❤♦✇ t❤❛t ✐❢ C n ❛♥❞

C ❛r❡ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥s ❛ss♦❝✐❛t❡❞ ✇✐t❤ F n , F ✱ ✇✐t❤ r❡s♣❡❝t✐✈❡ ♠❛r❣✐♥❛❧s G n ✱ G ✱ t❤❡♥✱

F n

→ d F ✐❢ ❛♥❞ ♦♥❧② ✐❢

✐✮ t❤❡ ♠❛r❣✐♥s ✇❡❛❦❧② ❝♦♥✈❡r❣❡✿ F n,j

→ d F j ✱ j = 1, . . . , d❀

✐✐✮ ❛♥❞ C n ❝♦♥✈❡r❣❡s ✉♥✐❢♦r♠❧② t♦ C ♦♥ RanG ✳

◗✉♦t✐♥❣ ▲✐♥❞♥❡r ❛♥❞ ❙③✐♠❛②❡r✱ ✏s✐♥❝❡ ✐♥ ❬t❤❡ ❞✐s❝r❡t❡❪ ❝❛s❡✱ t❤❡ ❝♦♣✉❧❛ ♦❢ X ❞♦❡s ♥♦t

♥❡❡❞ t♦ ❜❡ ✉♥✐q✉❡✱ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦♣✉❧❛s ♦♥ [0, 1] d ❝❛♥♥♦t ❜❡ ❡①♣❡❝t❡❞✑ ❛♥❞ t❤❡②

♣r❡s❡♥t ❛ ❝♦✉♥t❡r✲❡①❛♠♣❧❡✳ ❙❡❡ ❛❧s♦ ◆❡s❧❡❤♦✈❛ ❬✶✸❪ ❢♦r ❛ s✐♠✐❧❛r ❞✐s❝✉ss✐♦♥✳

❚❤❡ q✉❡st✐♦♥ t❤✉s ❛r✐s❡s t♦ ✇❤❛t ❝❛♥ ❜❡ ❡①♣❡❝t❡❞ ✇❤❡♥ ♦♥❡ ❛♣♣❧✐❡s t❤❡ ♣r❡✈✐♦✉s s❝❤❡♠❡s ♠❡♥t✐♦♥❡❞ ✐♥ s❡❝t✐♦♥ ✶✳✶ ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ s♦♠❡ s♣❡❝✐✜❝ ❣❡♥✉✐♥❡ ❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛

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

❡①tr❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω , A , P )✱

✐✮ ❧❡t X n ∼ F n ✳ ❙❡t U n := G n (X n , V) t❤❡ ♠✉❧t✐✈❛r✐❛t❡ ❞✐str✐❜✉t✐♦♥❛❧ tr❛♥s❢♦r♠s ❢♦r t❤❡ ❡❝❞❢ F n ✱ ✇✐t❤ r❛♥❞♦♠✐s❛t✐♦♥ V ✳ ❉❡♥♦t❡ ❛s C n 1 t❤❡ ❝❞❢ ♦❢ U n ✱ ✐✳❡✳ t❤❡ ❝♦♣✉❧❛

❢✉♥❝t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F n ❀

✐✐✮ ❧❡t X n ∼ F n ❛s ❜❡❢♦r❡ ❛♥❞ Z ∼ K ❛ ❝♦♥t✐♥✉♦✉s ✈❡❝t♦r✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ (X 1 , X 2 , . . .)✳

❙❡t

Y n := X n + h n Z, h n ↓ 0,

t❤❡ s♠♦♦t❤✐♥❣ ♦❢ X n ✳ ❉❡♥♦t❡ ❛s F ˆ n , G ˆ n t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✭❝♦♥t✐♥✉♦✉s✮ ❥♦✐♥t ❛♥❞

♠❛r❣✐♥❛❧ ❝❞❢s✳ ◆♦t❡ t❤❛t F ˆ n ✐s t❤❡ P❛r③❡♥✲❘♦s❡♥❜❧❛tt ❦❡r♥❡❧✲s♠♦♦t❤❡❞ ❡♠♣✐r✐❝❛❧

❝❞❢✱

P (Y n ≤ x) = Z

P

Z ≤ x − y h n

dF n (y) = 1 n

n

X

i=1

K

x − X i

h n

.

❉❡✜♥❡ U ˆ n := G ˆ n (Y n ), t❤❡ ❝♦♣✉❧❛ r❡♣r❡s❡♥t❡r ❛ss♦❝✐❛t❡❞ ✇✐t❤ F ˆ n ✱ ♦❜t❛✐♥❡❞ ✈✐❛

t❤✐s ♣r♦❜❛❜✐❧✐t② ✐♥t❡❣r❛❧ tr❛♥s❢♦r♠✳ ❉❡♥♦t❡ ❛s C n 2 t❤❡ ❝❞❢ ♦❢ U ˆ n ✱ ✐✳❡✳ t❤❡ ❝♦♣✉❧❛

❢✉♥❝t✐♦♥ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F ˆ n ✳

❋❛✉❣❡r❛s✬ ❬✻❪ t❤❡♦r❡♠s ✸✳✶ ❛♥❞ ✹✳✶ ❡①♣❧♦r❡ t❤❡ ❧❛tt❡r s❝❤❡♠❡✿ ❤❡ s❤♦✇s t❤❛t t❤❡

s❡q✉❡♥❝❡ ♦❢ ❝♦♣✉❧❛ r❡♣r❡s❡♥t❡rs U ˆ n ✱ ❜✉✐❧t ❢r♦♠ t❤❡ ❡♠♣✐r✐❝❛❧ ❝❞❢ ❜② ♣r♦❜❛❜✐❧✐st✐❝ ❝♦♥✲

t✐♥✉❛t✐♦♥✱ ❝♦♥✈❡r❣❡s t♦ s♦♠❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♣✉❧❛ r❡♣r❡s❡♥t❡r U ˆ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡

(5)

♦r✐❣✐♥❛❧ X ✳ ❲❤❡♥ X ❤❛s ❛ ❞✐s❝r❡t❡ ❝♦♠♣♦♥❡♥t✱ t❤❡♦r❡♠ ✹✳✶ ♦❢ ❬✻❪ ❣✐✈❡s✱ ✈✐❛ ❛ ❞♦✉❜❧❡

❛s②♠♣t♦t✐❝ ❛r❣✉♠❡♥t✱ t❤❡ ✉♥✐❢♦r♠ ❛✳s✳ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✭❦❡r♥❡❧ s♠♦♦t❤❡❞✮ ❡♠♣✐r✐❝❛❧

❝♦♣✉❧❛ ❢✉♥❝t✐♦♥ C n 2 ♦♥ t❤❡ ✇❤♦❧❡ [0, 1] d ✱ ❛s ❛ s✐♠♣❧❡ ❝♦r♦❧❧❛r②✳

❆t ✜rst s✐❣❤t✱ ❛ r❡❛❞❡r ♠✐❣❤t ❜❡ t❡♠♣t❡❞ t♦ t❤✐♥❦ t❤❛t s✉❝❤ ❛ ✉♥✐❢♦r♠ ❝♦♥✈❡r❣❡♥❝❡

r❡s✉❧t ♦❢ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥s✱ ✐♥ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✱ ♦♥ t❤❡ ✇❤♦❧❡ [0, 1] d ❛♥❞ ♥♦t s♦❧❡❧② ♦♥

RanG ✱ ✐s ✐♥ ❝♦♥tr❛❞✐❝t✐♦♥ ✇✐t❤ t❤❡ ❛ss❡rt✐♦♥s ♦❢ ▲✐♥❞♥❡r✱ ❙③✐♠❛②❡r ❛♥❞ ❉❡❤❡✉✈❡❧s✳ ❍♦✇✲

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

❉❡❤❡✉✈❡❧s ❛♥❞ ▲✐♥❞♥❡r ❛♥❞ ❙③✐♠❛②❡r✬s r❡s✉❧t ❛r❡ ❧❡❢t ✉♥s♣❡❝✐✜❡❞✱ ✇❤✐❝❤ ❧❡❛✈❡s s♦♠❡

s♣❛❝❡ ❢♦r ❝♦✉♥t❡r❡①❛♠♣❧❡s✳ ❚♦ t❤❡ ❝♦♥tr❛r②✱ t❤❡ ✐♥❞❡t❡r♠✐♥❛❝② ♦♥ t❤❡ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥s

✐s r❛✐s❡❞ ✐♥ t❤❡♦r❡♠ ✹✳✶ ♦❢ ❋❛✉❣❡r❛s ❬✻❪✱ ❜② t❤❡ s♣❡❝✐✜❝ ❝♦♥str✉❝t✐♦♥ ♦❢ U ˆ n , U ˆ ✐♥✈♦❧✈❡❞

t❤❡r❡✳

✶✳✹✳ ❖✉t❧✐♥❡

❚❤❡ ♣r❡s❡♥t ❛rt✐❝❧❡ ❣r❡❛t❧② ✐♠♣r♦✈❡s ❛♥❞ ❝♦♠♣❧❡t❡s t❤❡s❡ r❡s✉❧ts✱ ❜② ✐♥✈❡st✐❣❛t✐♥❣

t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛s C n 1 ❛♥❞ C n 2 ✱ ✇❤❡♥ t❤❡

♦r✐❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥ F ♦❢ X ✐s ♣✉r❡❧② ❞✐s❝r❡t❡✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ ♣r❡s❡♥t ✐♥ s❡❝t✐♦♥

✷ ❛ ♠❛①✐♠❛❧ ❝♦✉♣❧✐♥❣ ❝♦♥str✉❝t✐♦♥✱ ✐♥s♣✐r❡❞ ❜② ❚❤♦r✐ss♦♥ ❬✷✶❪✱ ✇❤✐❝❤ ✐s ❛♣♣❧✐❡❞ t♦ t❤❡

❡♠♣✐r✐❝❛❧ ❝❞❢ ♦❢ ❛ ♣✉r❡❧② ❞✐s❝r❡t❡ ❞✐str✐❜✉t✐♦♥✳ ❚❤✐s ♣r❡❧✐♠✐♥❛r② ❦❡② r❡s✉❧t✱ ✇❤✐❝❤ ✐s

♦❢ ✐♥❞❡♣❡♥❞❡♥t ✐♥t❡r❡st✱ ✐s ♦❜t❛✐♥❡❞ ✉♥❞❡r t❤❡ ✈❡r② ✇❡❛❦ ❛ss✉♠♣t✐♦♥ ♦❢ ❡r❣♦❞✐❝✐t② ♦❢

t❤❡ s❛♠♣❧❡ X 1 , X 2 , . . .✳ ❚❤✐s ❢r❛♠❡✇♦r❦ ❡♥❝♦♠♣❛ss❡s ♠♦st t②♣❡s ♦❢ s❡r✐❛❧ ❞❡♣❡♥❞❡♥❝❡

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

✇❤❡♥ t❤❡ s❛♠♣❧❡ ✐s ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❛♣❡r✐♦❞✐❝ ♣♦s✐t✐✈❡ r❡❝✉rr❡♥t ▼❛r❦♦✈ ❝❤❛✐♥ ✇❤✐❝❤ ♣♦ss❡s

❛♥ ✐♥✈❛r✐❛♥t ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✱ ✇❤❡♥ ✐t ✐s ❛ st❛t✐♦♥❛r② ❡r❣♦❞✐❝ ❞②♥❛♠✐❝❛❧ s②st❡♠✱

❛♥❞✱ ♦❢ ❝♦✉rs❡✱ t❤❡ ✐✳✐✳❞✳ ❝❛s❡✮✳

❘❡❣❛r❞✐♥❣ ❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛s✱ t❤❡ ✐♥t❡r❡st ♦❢ t❤✐s ❝♦♥str✉❝t✐♦♥ ✐s t✇♦❢♦❧❞✿ ❢♦r t❤❡

❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥ C n 2 ❞❡r✐✈❡❞ ❜② ♣r♦❜❛❜✐❧✐st✐❝ ❝♦♥t✐♥✉❛t✐♦♥✱ ✐t ❛❧❧♦✇s t♦ ♦❜t❛✐♥

❛ s✐♠♣❧❡ ♣r♦♦❢ ♦❢ ✉♥✐❢♦r♠ ❛✳s✳ ❝♦♥s✐st❡♥❝② ♦♥ [0, 1] d ♦❢ C n 2 ✱ ✇❤✐❝❤ ❜②♣❛ss❡s t❤❡ ❞♦✉❜❧❡

❛s②♠♣t♦t✐❝ ❢r❛♠❡✇♦r❦ ✐♥✈♦❧✈❡❞ ✐♥ t❤❡♦r❡♠ ✹✳✶ ♦❢ ❬✻❪✳ ▼♦r❡ ✐♠♣♦rt❛♥t❧②✱ ❢♦r t❤❡ ❡♠♣✐r✐❝❛❧

❝♦♣✉❧❛ ❢✉♥❝t✐♦♥ C n 1 ♦❜t❛✐♥❡❞ ✈✐❛ t❤❡ ❞✐str✐❜✉t✐♦♥❛❧ tr❛♥s❢♦r♠✱ ✇❡ ❛❧s♦ ♦❜t❛✐♥ t❤❡ ♥❡✇

r❡s✉❧t ♦❢ ❛✳s✳ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦♣✉❧❛ r❡♣r❡s❡♥t❡rs U n

a.s. → U ✱ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣

✉♥✐❢♦r♠ ❛✳s✳ ❝♦♥✈❡r❣❡♥❝❡ ♦♥ [0, 1] d ♦❢ C n 1 ✳ ❊✈❡♥t✉❛❧❧②✱ ✇❡ s❤♦✇✱ ✐♥ s❡❝t✐♦♥ ✹✱ ❤♦✇ t❤❡s❡

r❡s✉❧ts ❛❧❧♦✇ t♦ ♦❜t❛✐♥ ❛s s✐♠♣❧❡ ❝♦r♦❧❧❛r✐❡s ❝♦♥s✐st❡♥❝② ♦❢ s♦♠❡ ❡st✐♠❛t♦rs ♦❢ ❡①t❡♥s✐♦♥s

♦❢ ❝♦♥❝♦r❞❛♥❝❡ ♠❡❛s✉r❡s ❢♦r ❞✐s❝r❡t❡ ✈❡❝t♦rs✱ s✉❝❤ ❛s ◆❡s❧❡❤♦✈❛✬s ❬✶✸❪ ❛♥❞ ▼❡s✜♦✉✐ ❛♥❞

◗✉❡ss②✬s ❬✶✵❪ ❡①t❡♥s✐♦♥s ♦❢ ❑❡♥❞❛❧❧✬s t❛✉✳

✷✳ ▼❛①✐♠❛❧ ❝♦✉♣❧✐♥❣ ♦❢ t❤❡ ❞✐s❝r❡t❡ ❡♠♣✐r✐❝❛❧ ❝❞❢

✷✳✶✳ ❈♦✉♣❧✐♥❣ ♦❢ ❞✐s❝r❡t❡ r❛♥❞♦♠ ✈❡❝t♦rs

❈♦✉♣❧✐♥❣ ✐s ❛ ♣♦✇❡r❢✉❧ ♣r♦❜❛❜✐❧✐st✐❝ ♠❡t❤♦❞ ✇❤✐❝❤ ❛❧❧♦✇s t♦ t✉r♥ ❞✐str✐❜✉t✐♦♥❛❧ ♣r♦♣✲

❡rt✐❡s ✐♥t♦ t❤❡✐r ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦✉♥t❡r♣❛rts ✐♥ t❡r♠s ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ❙❡❡ ❡✳❣✳ ▲✐♥❞✲

✈❛❧❧ ❬✾❪ ❛♥❞ ❚❤♦r✐ss♦♥ ❬✷✶❪ ❢♦r s♦♠❡ ❡①❝❡❧❧❡♥t tr❡❛t✐s❡s ♦♥ t❤❡ s✉❜❥❡❝t✳ ❆s ✐♥ ❚❤♦r✐ss♦♥

❬✷✶❪✱ ❝❤❛♣t❡r ✶✱ r❡❝❛❧❧ t❤❛t ✐❢ {X i , i ∈ I } ❜❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ r❛♥❞♦♠ ✈❡❝t♦rs ♦♥ R d ✱ ✇❤❡r❡

I ✐s ❛♥ ✐♥❞❡① s❡t✱ t❤❡♥ ❛ ❝♦✉♣❧✐♥❣ ♦❢ {X i , i ∈ I } ✐s ❛ ❢❛♠✐❧② ♦❢ r❛♥❞♦♠ ✈❡❝t♦rs ( X ˆ i , i ∈ I )

❞❡✜♥❡❞ ♦♥ t❤❡ s❛♠❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✱ s✉❝❤ t❤❛t X i

= d X ˆ i ✱ ❢♦r ❛❧❧ i ∈ I ✳ X ˆ i ✐s ❝❛❧❧❡❞ ❛ r❡♣r❡s❡♥t❡r ♦r ❛ ❝♦♣② ♦❢ X i ✳ ❆s ❛ss❡rt❡❞ ❛❜♦✈❡✱ ✐t ✐s ✇❡❧❧ ❦♥♦✇♥ t❤❛t t❤❡ ❞✐str✐❜✉t✐♦♥❛❧

♣r♦♣❡rt② ♦❢ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♠❡❛s✉r❡s ❝❛♥ ❜❡ tr❡❛t❡❞ ✐♥ ❛ ♣r♦❜❛❜✐❧✐st✐❝ ♠❛♥♥❡r ❜②

(6)

❝♦✉♣❧✐♥❣ ♠❡t❤♦❞s✿ t❤❡ ❙❦♦r♦❦❤♦❞✲❉✉❞❧❡②✲❲✐❝❤✉r❛✬s t❤❡♦r❡♠ ✭s❡❡ ❙❦♦r♦❦❤♦❞ ❬✷✵❪✱ ❉✉❞✲

❧❡② ❬✺❪✮✱ ❛ss❡rts t❤❛t✱ ✐❢ X n ∼ F n , X ∼ F ✱ n ∈ N ✱ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ F n t♦✇❛r❞s F✱

F n

→ d F ✱ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ❝♦✉♣❧✐♥❣ ( X ˆ 1 , X ˆ 2 , . . . , X ˆ ) ♦❢ X 1 , X 2 , . . . , X ✱ s✉❝❤ t❤❛t X ˆ n a.s. → X ˆ ✳

■♥ t❤❡ ❝❛s❡ t❤❡ r❛♥❞♦♠ ✈❡❝t♦rs ✐♥✈♦❧✈❡❞ ❛r❡ ❛❧❧ ♣✉r❡❧② ❞✐s❝r❡t❡ ❛♥❞ t❛❦❡ t❤❡✐r ✈❛❧✉❡s

♦♥ ❛ ❝♦♠♠♦♥ ❝♦✉♥t❛❜❧❡ s❡t E✱ ❚❤♦r✐ss♦♥ ❬✷✶❪ s♦♠❡❤♦✇ str❡♥❣t❤❡♥s t❤✐s r❡s✉❧t✿ ♣♦✐♥t✇✐s❡

❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥s ♦❢ r❛♥❞♦♠ ✈❡❝t♦rs ❝❛♥ ❜❡ t✉r♥❡❞ ✐♥t♦ ♣♦✐♥t✲

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

❝♦♥str✉❝t✐♦♥✳

❚❤❡♦r❡♠ ✷✳✶ ✭❚❤♦r✐ss♦♥ t❤❡♦r❡♠ ✶✳✻✳✶✮✳ ▲❡t X 1 , X 2 , . . . , X ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ❞✐s❝r❡t❡

r❛♥❞♦♠ ✈❡❝t♦rs ✇✐t❤ ✈❛❧✉❡s ♦♥ ❛ ❝♦♠♠♦♥ ❝♦✉♥t❛❜❧❡ s❡t E✳ ❚❤❡♥✱

n→∞ lim P ( X n = x ) = P( X = x ), ❢♦r ❛❧❧ x ∈ E,

❤♦❧❞s ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝♦✉♣❧✐♥❣ ( X ˆ 1 , . . . , X ˆ ∞ ) ♦❢ X 1 , . . . , X ∞ ❛♥❞ ❛ ✜♥✐t❡

r❛♥❞♦♠ ✐♥t❡❣❡r N s✳t✳

X ˆ n = X ˆ ∞ , n ≥ N .

Pr♦♦❢✳ ❙❡❡ ❛♣♣❡♥❞✐①✱ ✇❤❡r❡ ❚❤♦r✐ss♦♥✬s ♣r♦♦❢ ✐s r❡♣r♦❞✉❝❡❞ ❢♦r ❝♦♥✈❡♥✐❡♥❝❡✳ ❍❡ ♣r♦✈❡❞

t❤❡ r❡s✉❧t ❢♦r r❡❛❧ ✈❛❧✉❡❞ r❛♥❞♦♠✲✈❛r✐❛❜❧❡s ❜✉t ✐t ❡①t❡♥❞s ❡✛♦rt❧❡ss❧② t♦ r❛♥❞♦♠ ✈❡❝t♦rs✳

❘❡♠❛r❦ ✶✳ ❚❤✐s r❡s✉❧t ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s ❛ st♦❝❤❛st✐❝ ❛♥❛❧♦❣ ♦❢ t❤❡ ❢❛❝t t❤❛t ✐❢ (u n ) ✐s

❛ ❞❡t❡r♠✐♥✐st✐❝ s❡q✉❡♥❝❡ ✇✐t❤ ✈❛❧✉❡s ✐♥✱ s❛②✱ Z ✱ t❤❡♥ (u n ) ❝♦♥✈❡r❣❡s ✐✛ (u n ) ✐s ❡✈❡♥t✉❛❧❧② st❛t✐♦♥❛r② ✐♥ t❤❡ s❡♥s❡ t❤❛t t❤❡r❡ ❡①✐sts p ∈ N ✱ s✳t✳ ∀n ≥ p, u n+1 = u n ✳

❘❡♠❛r❦ ✷✳ ◆♦t❡ t❤❛t ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥s ✐s ♦❜✈✐♦✉s❧② str♦♥❣❡r t❤❛♥ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡✿ ❢♦r X n = 1/n✱ n ∈ N ❛♥❞ X ∞ = 0✱ ❝❧❡❛r❧② X n

→ d X ∞ ✱

✇❤❡r❡❛s P(X n = 0) = 0✱ ❢♦r ❛❧❧ n ∈ N ✱ ❛♥❞ P (X ∞ = 0) = 1✳ ❚❤❡r❡❢♦r❡✱ t❤❡♦r❡♠ ✷✳✶

✐s ♥♦t✱ str✐❝t❧② s♣❡❛❦✐♥❣✱ ❛ str❡♥❣t❤❡♥✐♥❣ ♦❢ ❙❦♦r♦❦❤♦❞✬s t❤❡♦r❡♠✳ ❙❡❡ ❚❤♦r✐ss♦♥✱ ❬✷✶❪

❝❤❛♣t❡r ✶ ❢♦r ❛ ❞✐s❝✉ss✐♦♥✳

✷✳✷✳ ▼❛①✐♠❛❧ ❝♦✉♣❧✐♥❣ ♦❢ t❤❡ ❡♠♣✐r✐❝❛❧ ❝❞❢ ❢♦r ❞✐s❝r❡t❡ ✈❡❝t♦rs

▲❡t µ ❛ ♠✉❧t✐✈❛r✐❛t❡ ❛t♦♠✐❝ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡✱ ✐✳❡✳ ❝❤❛r❣✐♥❣ ❛ ❝♦✉♥t❛❜❧❡ s❡t E = {x 1 , x 2 , . . .} ♦❢ ✈❛❧✉❡s✱ ✇✐t❤ ❝❞❢ F ✳ ❉❡♥♦t❡ ✐ts ♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥ ❜② p(x)✿ p(x) > 0✱

∀x ∈ E ❛♥❞ P

x ∈E p(x) = 1✳ ❆ss✉♠❡ t❤❛t X 1 , X 2 , . . . , ❛♥ ❡r❣♦❞✐❝ s❛♠♣❧❡ ♦❢ µ✱ ✐♥ t❤❡

s❡♥s❡ t❤❛t✱ X 1 , X 2 , . . . , t❛❦❡ t❤❡✐r ✈❛❧✉❡s ✐♥ E✱ ❛♥❞✱ ❢♦r ❛❧❧ x ∈ E✱

1 n

n

X

i=1

1 X

i

= x

a.s. → p( x ), ❛s n → ∞. ✭✷✮

❉❡♥♦t❡ F n t❤❡ ❡♠♣✐r✐❝❛❧ ❝❞❢ ❝♦rr❡♣♦♥❞✐♥❣ t♦ p n ✳ ❖♥❡ t❤❡♥ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❦❡② ♣r♦♣♦✲

s✐t✐♦♥✿

Pr♦♣♦s✐t✐♦♥ ✷✳✷✳ ▲❡t X 1 , X 2 , . . . , ❛♥ ❡r❣♦❞✐❝ s❛♠♣❧❡ ♦❢ ♣✉r❡❧② ❞✐s❝r❡t❡ r❛♥❞♦♠ ✈❡❝t♦rs

❞✐str✐❜✉t❡❞ ❛s F✱ s✳t✳ ✭✷✮ ✐s s❛t✐s✜❡❞✳ ❚❤❡♥✱ t❤❡r❡ ❡①✐sts s♦♠❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ s✳t✳ ♦♥❡

❝❛♥ ❞❡✜♥❡ ❥♦✐♥t❧② ♦♥ ✐t

(7)

✐✮ ❛ s❡q✉❡♥❝❡ X ˆ n ∼ F n ,

✐✐✮ X ˆ ∼ F✱

✐✐✐✮ ❛ ✜♥✐t❡ r❛♥❞♦♠ ✐♥t❡❣❡r N ✱

s✳t✳ X ˆ n = X ˆ , ❢♦r n ≥ N , ❝♦♥❞✐t✐♦♥❛❧❧② ♦♥ X 1 , X 2 , . . .✱ ❢♦r ❛❧♠♦st ❡✈❡r② s❡q✉❡♥❝❡

X 1 , X 2 , . . .✳

Pr♦♦❢✳ ❆ss✉♠❡ t❤❡ s❛♠♣❧❡ X 1 , X 2 , . . . , ✐s ❞❡✜♥❡❞ ♦♥ s♦♠❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω, A, P )✳

❋♦r ❡①❛♠♣❧❡✱ t❛❦❡ (Ω, A) t❤❡ ✐♥✜♥✐t❡ ❝♦✉♥t❛❜❧❡ ♣r♦❞✉❝t (S N , S N ) = (S×S×. . . , S×S×. . .)

✇❤❡r❡ S = R d ❛♥❞ S = B(S) t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❇♦r❡❧ σ−✜❡❧❞ ❛♥❞ ❧❡t X i (ω) : (S N , S N ) 7→

(S, S )✱ i = 1, 2, . . .✱ ❜❡ t❤❡ ❝♦♦r❞✐♥❛t❡✲♣r♦❥❡❝t✐♦♥ ♠❛♣♣✐♥❣✱ ❞❡✜♥❡❞ ❢♦r ω = (ω 1 , ω 2 , . . .) ∈ S N ✱ ❜② X i (ω) = ω i ✳

❈❛❧❧ P n ω (.) = n 1 P

δ X

i

(ω) (.) t❤❡ ❡♠♣✐r✐❝❛❧ ♠❡❛s✉r❡ ❜❛s❡❞ ♦♥ t❤❡ s❛♠♣❧❡✳ ❉❡♥♦t❡

p n (x) = P n ω ({x}) t❤❡ ❡♠♣✐r✐❝❛❧ ♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥✱ ✭✇❤❡r❡ ✇❡ s✉♣♣r❡ss❡❞ t❤❡

❞❡♣❡♥❞❡♥❝❡ ♦♥ ω✮✳ ❆ss✉♠♣t✐♦♥ ✭✷✮ ♠❡❛♥s t❤❛t t❤❡r❡ ❡①✐sts s♦♠❡ s❡t Ω 0 ⊂ Ω ✇✐t❤

P(Ω 0 ) = 1 s✳t✳ ❢♦r ❛❧❧ ω ∈ Ω 0 ✱ ❢♦r ❛❧❧ x ∈ E✱ p n (x) → p(x) ✳

❋✐① ω ∈ Ω 0 ✳ ❖♥ s♦♠❡ ❡①tr❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω , A , P )✱ ❞❡✜♥❡ ❛ s❡q✉❡♥❝❡ X n P

F n ♦❢ r❛♥❞♦♠ ✈❡❝t♦rs ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ F n ❛♥❞ X P

F ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦

F✳ ❊✈❡♥ t❤♦✉❣❤ t❤❡ s❡t ♦❢ ✈❛❧✉❡s ♦❢ X n ∼ F n ✐s ✜♥✐t❡ ❛♥❞ ✐♥❝❧✉❞❡❞ ✐♥ E✱ ❤❡♥❝❡ ♣♦ss✐❜❧②

♥♦t ❡q✉❛❧ t♦ E✱ ✐t ✐s ♥♦t ❞✐✣❝✉❧t t♦ s❡❡ t❤❛t t❤❡♦r❡♠ ✷✳✶ r❡♠❛✐♥s ✈❛❧✐❞ ✐♥ t❤✐s s❡tt✐♥❣✿

s✐♥❝❡ t❤❡ ❡♠♣✐r✐❝❛❧ ♠❡❛s✉r❡s (P n ω ) ❛r❡ ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ✇✳r✳t✳ t♦ ❛ s✐♥❣❧❡ ❞♦♠✐♥❛t✐♥❣

♠❡❛s✉r❡ µ✱ ♦♥❡ ❝❛♥ ✉s❡ ❚❤♦r✐ss♦♥ ❬✷✶❪ t❤❡♦r❡♠ ✼✳✶ ✐♥ ❝❤❛♣t❡r ✶ ✭s❡❡ ❛❧s♦ ❤✐s t❤❡♦r❡♠ ✾✳✷

✐♥ ❝❤❛♣t❡r ✸✮ ✐♥st❡❛❞ ♦❢ ❤✐s t❤❡♦r❡♠ ✶✳✻✳✶✱ ✇✐t❤ ❘❛❞♦♥✲◆✐❦♦❞②♠ ❞❡r✐✈❛t✐✈❡s ✐♥st❡❛❞ ♦❢

♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥s✳

❖♥❡ ❝❛♥ t❤❡r❡❢♦r❡ ❛♣♣❧② t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡♦r❡♠ ✷✳✶✿ t❤❡r❡ ❡①✐ts ❛ ❝♦✉♣❧✐♥❣

( X ˆ 1 , X ˆ 2 , . . . , X ˆ ) ♦❢ X 1 , X 2 , . . . , X ❛♥❞ ❛ ✜♥✐t❡ r❛♥❞♦♠ ✐♥t❡❣❡r N s✳t✳

X ˆ n = X ˆ , n ≥ N .

❚❤❡ ❝♦♥str✉❝t✐♦♥ ✐s ✈❛❧✐❞ ❢♦r ❡✈❡r② ω ∈ Ω 0 ✱ ✇✐t❤ P (Ω 0 ) = 1✱ ❤❡♥❝❡ t❤❡ ❛ss❡rt✐♦♥✳

✸✳ ❯♥✐❢♦r♠ ❛✳s✳ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s♦♠❡ ❣❡♥✉✐♥❡ ❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥s ❢♦r

♣✉r❡❧② ❞✐s❝r❡t❡ ❞❛t❛

▲❡t X ∈ R d ❜❡ ❛ ♣✉r❡❧② ❞✐s❝r❡t❡✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❡❝t♦r ✇✐t❤ ❝❞❢ F ✱ ♠❛r❣✐♥❛❧ ❝❞❢s G ✳

▲❡t X 1 , . . . , X n ❜❡ ❛♥ ❡r❣♦❞✐❝ s❛♠♣❧❡ ♦❢ X ✐♥ t❤❡ s❡♥s❡ ♦❢ ✭✷✮ ❛♥❞ F n , G n t❤❡ ❡♠♣✐r✐❝❛❧

❝❞❢s ❛s ❜❡❢♦r❡✳ ❚❤❡ ♣r❡❧✐♠✐♥❛r② ❝♦♥str✉❝t✐♦♥ ♦❢ s❡❝t✐♦♥ ✷ ❛❧❧♦✇s t♦ ♦❜t❛✐♥ s✐♠♣❧❡ ♣r♦♦❢s ♦❢

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

✸✳✶✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❞✐str✐❜✉t✐♦♥❛❧❧② tr❛♥s❢♦r♠❡❞ ❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛

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

✐s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ♥❡①t t❤❡♦r❡♠✳

❚❤❡♦r❡♠ ✸✳✶✳ ❋♦r F ♣✉r❡❧② ❞✐s❝r❡t❡✱ t❤❡r❡ ❡①✐sts ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✇❤✐❝❤ ❝❛rr✐❡s ❝♦♣✉❧❛

r❡♣r❡s❡♥t❡rs U ˆ n ∼ C n 1 , U ˆ ∼ C✱ s✉❝❤ t❤❛t✿

✐✮ C n 1 , C ❛r❡ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥s ❛ss♦❝✐❛t❡❞ ✇✐t❤ F n ✱ F ✱ r❡s♣❡❝t✐✈❡❧②❀

(8)

✐✐✮ U ˆ n

a.s. → U ˆ ✱ ❝♦♥❞✐t✐♦♥❛❧❧② ♦♥ X 1 , X 2 , . . .✱ ❢♦r ❛❧♠♦st ❡✈❡r② s❡q✉❡♥❝❡ X 1 , X 2 , . . .✳

Pr♦♦❢✳ ▲❡t X ∼ F, (X n ∼ F n ) n∈ N , V ❞❡✜♥❡❞ ♦♥ t❤❡ s❛♠❡ ♣r♦❞✉❝t ♣r♦❜❛❜✐❧✐t② s♣❛❝❡

✇✐t❤ V ❛ ✈❡❝t♦r ✇✐t❤ ✉♥✐❢♦r♠ ♠❛r❣✐♥❛❧s✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ (X 1 , X 2 , . . . , X )✳ ❙❡t U :=

G(X , V)✱ U n := G n (X n , V)✱ t❤❡ ♠✉❧t✐✈❛r✐❛t❡ ❞✐str✐❜✉t✐♦♥❛❧ tr❛♥s❢♦r♠s ❢♦r t❤❡ ❝❞❢ F

❛♥❞ t❤❡ ❡❝❞❢ F n ✱ ✇✐t❤ t❤❡ s❛♠❡ r❛♥❞♦♠✐③❛t✐♦♥ V ✳ ❇② ❘üs❝❤❡♥❞♦r❢✬s ❬✶✻❪✱ U, U n ❛r❡

✈❡❝t♦rs ✇✐t❤ ✉♥✐❢♦r♠ ♠❛r❣✐♥❛❧s ✇❤♦s❡ ❝❞❢ C, C n 1 ❛r❡ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥s ❛♥❞ s❛t✐s❢② ❙❦❧❛r✬s t❤❡♦r❡♠ ❢♦r F, F n ✱ r❡s♣❡❝t✐✈❡❧②✳

❇② ♣r♦♣♦s✐t✐♦♥ ✷✳✷✱ t❤❡r❡ ❡①✐sts ❛ ❝♦✉♣❧✐♥❣ ( X ˆ 1 , X ˆ 2 , . . . , X ˆ ) ♦❢ X 1 , X 2 , . . . , X ❛♥❞ ❛

✜♥✐t❡ r❛♥❞♦♠ ✐♥t❡❣❡r N s✉❝❤ t❤❛t

✐✮ X ˆ n = d X n ✱ X ˆ = d X ✱ n ≥ 1✱

✐✐✮ X ˆ n = X ˆ , n ≥ N ✳

❊♥❧❛r❣❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✐❢ ♥❡❝❡ss❛r② t♦ ❝❛rr② s♦♠❡ ✈❡❝t♦r V ˆ = d V ✱

✐♥❞❡♣❡♥❞❡♥t ♦❢ ( X ˆ 1 , X ˆ 2 , . . . , X ˆ , N )✳ ❚❤✉s✱ U n = G n (X n , V) = d U ˆ n := G n ( X ˆ n , V) = ˆ G n ( X ˆ , V) ˆ ❢♦r n ≥ N ✳ p n (x) → p(x) ❡♥t❛✐❧s G n (x, λ) → G(x, λ) ♣♦✐♥t✇✐s❡✳ ■♥ t✉r♥✱ ❛s n → ∞✱ s✐♥❝❡ N ✐s ✜♥✐t❡✱

U ˆ n a.s. → U ˆ := G( X ˆ , V) ˆ = d G(X , V) = U.

❘❡♠❛r❦ ✸✳ ❚❤❡ ❢❛❝t t❤❛t G ✐s ♥♦t ❝♦♥t✐♥✉♦✉s ♣r❡✈❡♥ts ♦♥❡ t♦ ♠✐♠✐❝ t❤❡ ♣r♦♦❢ ❜②

❋❛✉❣❡r❛s ❬✻❪ t❤❡♦r❡♠ ✹✳✶✱ ✐✳❡✳ t♦ ✉s❡ ❙❦♦r♦❦❤♦❞✬s r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r❡♠ ❛♥❞ t❤❡ ❝♦♥t✐♥✲

✉♦✉s ♠❛♣♣✐♥❣ t❤❡♦r❡♠ ❞✐r❡❝t❧② t♦ G n (X n , V)✳

❈♦r♦❧❧❛r② ✸✳✷✳ ❲✐t❤ P ✲♣r♦❜❛❜✐❧✐t② ♦♥❡✱

sup

u ∈[0,1]

d

|C n 1 (u) − C(u)| → 0.

Pr♦♦❢✳ ❚❤❡♦r❡♠ ✸✳✶ ✐♠♣❧✐❡s U ˆ n

→ d U ˆ ✳ ❙✐♥❝❡ C ✐s ❝♦♥t✐♥✉♦✉s✱ P♦❧②❛✬s ❧❡♠♠❛ ❡♥t❛✐❧s t❤❡ ❞❡s✐r❡❞ ✉♥✐❢♦r♠ ❝♦♥s✐st❡♥❝② ♦❢ ❝❞❢s✳ ❙✐♥❝❡ t❤❡ r❡s✉❧t ✐s ✈❛❧✐❞ ❢♦r ❛❧❧ ω ∈ Ω 0 ✇✐t❤

P(Ω 0 ) = 1✱ t❤❡ ❛ss❡rt✐♦♥ ♦❝❝✉rs✳

✸✳✷✳ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ ❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛s r❡❣✉❧❛r✐s❡❞ ❜② ❝♦♥✈♦❧✉t✐♦♥

❚❤❡ ❛✳s✳ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❦❡r♥❡❧✲s♠♦♦t❤❡❞ ❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛ r❡♣r❡s❡♥t❡r ✐s ♣r❡s❡♥t❡❞

✐♥ t❤❡ ♥❡①t t❤❡♦r❡♠✳

❚❤❡♦r❡♠ ✸✳✸✳ ❋♦r F ♣✉r❡❧② ❞✐s❝r❡t❡✱ t❤❡r❡ ❡①✐sts ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✇❤✐❝❤ ❝❛rr✐❡s ❝♦♣✉❧❛

r❡♣r❡s❡♥t❡rs U ˆ n ∼ C n 2 , U ˆ ∼ C r❡s♣❡❝t✐✈❡❧②✱ s✉❝❤ t❤❛t✿

✐✮ C n 2 , C ❛r❡ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥s✱ C ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ F✱ C n 2 ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❦❡r♥❡❧✲

s♠♦♦t❤❡❞ ❡♠♣✐r✐❝❛❧ ❝❞❢❀

✐✐✮ U ˆ n

→ d U ˆ ✱ ❢♦r ❛ s✉❜s❡q✉❡♥❝❡ n → ∞✱ ❝♦♥❞✐t✐♦♥❛❧❧② ♦♥ X 1 , X 2 , . . .✱ ❢♦r ❛❧♠♦st

❡✈❡r② s❡q✉❡♥❝❡ X 1 , X 2 , . . .✳

(9)

Pr♦♦❢✳ ❉❡♥♦t❡ ❛s ❜❡❢♦r❡ ( X ˆ 1 , X ˆ 2 , . . . , X ˆ ) ❚❤♦r✐ss♦♥✬s ❝♦✉♣❧✐♥❣ ♦❢ ♦❢ X 1 , X 2 , . . . , X ❛♥❞

N t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❛♥❞♦♠ ✜♥✐t❡ ✐♥t❡❣❡r s✉❝❤ t❤❛t X ˆ n = X ˆ ❢♦r n ≥ N ✱ ✇✐t❤ X ˆ n ∼ F n ✱ X ˆ ∼ F ✳ ❊♥❧❛r❣❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✐❢ ♥❡❝❡ss❛r② t♦ ❝❛rr② ❛ ❝♦♥t✐♥✉♦✉s

✈❡❝t♦r Z ∼ K✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ( X ˆ 1 , X ˆ 2 , . . . , X ˆ )✳ ❙❡t Y n := X ˆ n + h n Z, h n ↓ 0,

t❤❡ s♠♦♦t❤✐♥❣ ♦❢ X ˆ n ❀ ❞❡♥♦t❡ ❛s F ˆ n , G ˆ n t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❥♦✐♥t ❛♥❞ ♠❛r❣✐♥❛❧ ❝❞❢s ❛♥❞

s❡t U ˆ n := G ˆ n ( Y n )

t❤❡ ❝♦♣✉❧❛ r❡♣r❡s❡♥t❡r✳ F ˆ n ✐s t❤❡ ❦❡r♥❡❧✲s♠♦♦t❤❡❞ ❡♠♣✐r✐❝❛❧ ❝❞❢✳ ❙❡t X ¯ n = X ˆ + h n Z,

❞❡♥♦t❡ ❛s F ¯ n , G ¯ n t❤❡ ❥♦✐♥t ❛♥❞ ✈❡❝t♦r ♦❢ ♠❛r❣✐♥❛❧ ❝❞❢s ♦❢ X ¯ n ✱ ❛♥❞ ❧❡t U n := G ¯ n ( X ¯ n ).

❋♦r n ≥ N ✱ Y n = X ˆ + h n Z = X ¯ n ✳ ❚❤✉s✱

U ˆ n − U n = G ˆ n ( X ¯ n ) − G ¯ n ( X ¯ n ), ❢♦r n ≥ N .

❇✉t || G ˆ n − G ¯ n || ∞ ≤ ||G n − G|| ∞ → 0✱ ❛s n → ∞ ✱ s✐♥❝❡ ♣♦✐♥t✇✐s❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♣r♦❜❛✲

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

✭s❡❡ ❚❤♦r✐ss♦♥ ❬✷✶❪ s❡❝t✐♦♥ ✶✳✻✳✶✮✳ ❙✐♥❝❡ N ✐s ✜♥✐t❡✱ ❧❡tt✐♥❣ n → ∞ t❤❡r❡❢♦r❡ ❡♥t❛✐❧s

|| U ˆ n − U n || 1 ≤ ||||G n − G|| ∞ || 1 a.s. → 0.

❋❛✉❣❡r❛s ❬✻❪ t❤❡♦r❡♠ ✶✳✶✱ ❡♥t❛✐❧s U n

→ d U ✱ ❢♦r s♦♠❡ s✉❜s❡q✉❡♥❝❡ n → ∞ ❛♥❞ s♦♠❡ U ✱

✇❤♦s❡ ❝❞❢ ✐s ❛ ❝♦♣✉❧❛ ❛♥❞ s❛t✐s❢② ❙❦❧❛r✬s t❤❡♦r❡♠ ❛ss♦❝✐❛t❡❞ ✇✐t❤ F ✳ ❙❧✉ts❦②✬s t❤❡♦r❡♠

②✐❡❧❞s t❤❡ ❞❡s✐r❡❞ r❡s✉❧t✳

❈♦r♦❧❧❛r② ✸✳✹✳ ❲✐t❤ P ✲♣r♦❜❛❜✐❧✐t② ♦♥❡✱

sup

u ∈[0,1]

d

|C n 2

(u) − C(u)| → 0.

Pr♦♦❢✳ ❆s ✐♥ ❝♦r♦❧❧❛r② ✸✳✷✳

✹✳ ❈♦♥s✐st❡♥❝② ♦❢ ❝♦♥❝♦r❞❛♥❝❡ ♠❡❛s✉r❡ ❢♦r ❞✐s❝r❡t❡ ✈❡❝t♦rs

❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡♦r❡♠ ✸✳✶✱ ♦♥❡ ❛✉t♦♠❛t✐❝❛❧❧② ♦❜t❛✐♥s str♦♥❣ ❝♦♥s✐st❡♥❝② ♦❢ ❛♥②

❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥❛❧ ♦❢ U ❜② t❤❡ ❝♦♥t✐♥✉♦✉s ♠❛♣♣✐♥❣ t❤❡♦r❡♠✳ ❋♦r ❡①❛♠♣❧❡✱ ♦♥❡ ❝❛♥

♦❜t❛✐♥ str♦♥❣ ❝♦♥s✐st❡♥❝② ♦❢ ❝♦♥❝♦r❞❛♥❝❡ ♠❡❛s✉r❡s ❢♦r ❞✐s❝r❡t❡ ✈❡❝t♦rs✱ ❛s ✐s s❤♦✇♥ ❜❡❧♦✇✳

❲❡ ❢♦❝✉s ♠❛✐♥❧② ♦♥ ❑❡♥❞❛❧❧✬s τ ✱ ❜✉t s✐♠✐❧❛r r❡s✉❧ts ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢♦r ❙♣❡❛r♠❛♥✬s ρ✳

(10)

✹✳✶✳ ❈♦♥❝♦r❞❛♥❝❡ ♠❡❛s✉r❡ ❢♦r ❜✐✈❛r✐❛t❡ ❝♦♥t✐♥✉♦✉s ✈❡❝t♦rs

▲❡t X = (X, Y ) ❛ ❜✐✈❛r✐❛t❡ ✈❡❝t♦r ✇✐t❤ ❝❞❢ F ✳ ❚❤❡ ✐❞❡❛ ♦❢ ❝♦♥❝♦r❞❛♥❝❡ ✐s t❤❛t ❛♥

✐♥❝r❡❛s❡ ✭r❡s♣✳ ❛ ❞❡❝r❡❛s❡✮ ✐♥ X ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛♥ ✐♥❝r❡❛s❡ ✭r❡s♣✳ ❛ ❞❡❝r❡❛s❡✮ ✐♥

Y ✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❧❡t X = (X , Y ) ❛♥ ✐♥❞❡♣❡♥❞❡♥t ❝♦♣② ♦❢ X ✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢

❝♦♥❝♦r❞❛♥❝❡ ✐s ❞❡✜♥❡❞ ❛s

Q(X, X ) := P((X − X )(Y − Y ) > 0),

❛♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞✐s❝♦r❞❛♥❝❡ ❛s

Q(X, ¯ X ) := P((X − X )(Y − Y ) < 0).

❑❡♥❞❛❧❧✬s τ ❝♦❡✣❝✐❡♥t ✐s ❞❡✜♥❡❞ ❛s τ(F ) := Q(X, X )✱ ✇❤❡r❡

Q(X, X ) := Q(X, X ) − Q(X, ¯ X )

✐s ❝❛❧❧❡❞ t❤❡ ❝♦♥❝♦r❞❛♥❝❡ ❢✉♥❝t✐♦♥✱ s❡❡ ◆❡❧s❡♥ ❬✶✷❪✳

■❢ F ✐s ❝♦♥t✐♥✉♦✉s✱ P(tie) := P ((X − X )(Y − Y ) = 0) = 0✱ t❤✉s τ(F) = 2Q(X, X ) − 1 = 4

Z

C(u)dC(u) − 1 = 2Q(U, U ) − 1

✇❤❡r❡ U := G ( X ) ∼ C := F ◦ G −1 ✐s t❤❡ ✉♥✐q✉❡ ❝♦♣✉❧❛ r❡♣r❡s❡♥t❡r ❛ss♦❝✐❛t❡❞ t♦ X ✱

❛♥❞ s✐♠✐❧❛r❧② ❢♦r U = G ( X ) ∼ C = C✱ s❡❡ ◆❡❧s❡♥ ❬✶✷❪✳ ❍❡♥❝❡✱ ✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✱

τ(F) = τ(C)✱ ✐✳❡ t❤❡ ❝♦❡✣❝✐❡♥t ✐s t❤❡ s❛♠❡ ✇❤❡t❤❡r ♦♥❡ ❝♦♠♣✉t❡ ✐t ❛t t❤❡ ♦❜s❡r✈❛t✐♦♥❛❧

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

✐♥✈❛r✐❛♥t ✇✳r✳t✳ ❛♥② str✐❝t❧② ✐♥❝r❡❛s✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✳ ❍❡♥❝❡✱ ✐t ✐s s❝❛❧❡

✐♥✈❛r✐❛♥t ✐♥ t❤❡ ✉t♠♦st ♠❛♥♥❡r✱ ✇❤✐❝❤ ♠❛❦❡s ✐t ❛♥ ❛ttr❛❝t✐✈❡ ♠❡❛s✉r❡ ♦❢ ❞❡♣❡♥❞❡♥❝❡✳

✹✳✷✳ ❈♦♥❝♦r❞❛♥❝❡ ♠❡❛s✉r❡s ❢♦r ❜✐✈❛r✐❛t❡ ❞✐s❝r❡t❡ ✈❡❝t♦rs

■♥ t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✱ P (tie) 6= 0 ❛♥❞ t❤❡r❡ ❛r❡ ♠❛♥② ❝♦♣✉❧❛s ❛ss♦❝✐❛t❡❞ t♦ X ✱ ❛s

❡①♣❧❛✐♥❡❞ ✐♥ s❡❝t✐♦♥ ✶✳✶✳ ❚❤❡r❡❢♦r❡✱ ❡①♣r❡ss✐♥❣ ❛ ♣♦♣✉❧❛t✐♦♥ ✈❡rs✐♦♥ ♦❢ ❑❡♥❞❛❧❧✬s τ ✐♥

t❡r♠s ♦❢ ❛ ❝♦♣✉❧❛ ❢✉♥❝t✐♦♥ ❛ss♦❝✐❛t❡❞ t♦ X ✐s ♥♦ ❧♦♥❣❡r str❛✐❣❤t❢♦r✇❛r❞✳ ❉❡♥✉✐t ❛♥❞

▲❛♠❜❡rt ❬✹❪✱ ◆❡s❧❡❤♦✈❛ ❬✶✸❪ ❛♥❞ ▼❡s✜♦✉✐ ❛♥❞ ◗✉❡ss② ❬✶✵❪ ♣r♦♣♦s❡ ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢

❝♦♥❝♦r❞❛♥❝❡ ❝♦❡✣❝✐❡♥t ❢♦r ❞✐s❝r❡t❡ ✈❡❝t♦rs✱ ❜❛s❡❞ ♦♥ t❤❡ ❞✐str✐❜✉t✐♦♥❛❧ tr❛♥s❢♦r♠✳

▲❡t V := (V X , V Y ) ∼ R ❛ ❜✐✈❛r✐❛t❡ ✈❡❝t♦r ✇✐t❤ ✉♥✐❢♦r♠ [0, 1] ♠❛r❣✐♥❛❧s✳ ▲❡t V ❛♥

✐♥❞❡♣❡♥❞❡♥t ❝♦♣② ♦❢ V ✱ ✇✐t❤ ( V , V ) ❛❧s♦ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ( X , X )✳ ❙❡t U = G ( X , V )

✭r❡s♣✳ U = G ( X , V )✮ t❤❡ ❝♦♣✉❧❛ r❡♣r❡s❡♥t❡r ❛ss♦❝✐❛t❡❞ t♦ X ✭r❡s♣✳ X ✮ ✇✐t❤ r❛♥❞♦♠✲

✐③❡r V ✭r❡s♣ V ✮✳ ❬✶✵✱ ✹❪ ♣r♦♣♦s❡ t♦ t❛❦❡ ❛ r❛♥❞♦♠✐③❡r ✇✐t❤ ✐♥❞❡♣❡♥❞❡♥t ♠❛r❣✐♥❛❧s✱ ✐✳❡✳

R = Π t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ✭❝♦♣✉❧❛✮ ❞✐str✐❜✉t✐♦♥✱ ❛♥❞ t♦ ❞❡✜♥❡ ❦❡♥❞❛❧❧✬s τ ✐♥ t❡r♠s ♦❢ t❤❡

♣r♦❜❛❜✐❧✐t② ♦❢ ❝♦♥❝♦r❞❛♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♣✉❧❛ r❡♣r❡s❡♥t❡r✱ ✐✳❡✳

τ(F ) := Q(U, U ) = 2Q(U, U ) − 1,

s❡❡ ▼❡s✜♦✉✐ ❛♥❞ ◗✉❡ss② ❬✶✵❪ s❡❝t✐♦♥ ✹✳ ✭❙♦♠❡ ♠♦❞✐✜❡❞ ✈❡rs✐♦♥s ❜❛s❡❞ ♦♥ ♥♦r♠❛❧✐s❛t✐♦♥s

❛r❡ ❛❧s♦ ♣r♦♣♦s❡❞ ✐♥ ▼❡s✜♦✉✐ ❛♥❞ ◗✉❡ss② ❬✶✵❪ ❛♥❞ ◆❡s❧❡❤♦✈❛ ❬✶✸❪✮✳ ❚❤✐s ❛♣♣r♦❛❝❤ ✐s

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

❢✉♥❝t✐♦♥✱ s❡❡ ◆❡s❧❡❤♦✈❛ ❬✶✸❪ t❤❡♦r❡♠ ✺✱

Q(X, X ) = Q(U, U ) = 2Q(U, U ) − 1,

(11)

s♦ t❤❛t✱ ✐❢ ♦♥❡ s❡t C ❛s t❤❡ ❝❞❢ ♦❢ U ✱ t❤❡ ❝♦♥❝♦r❞❛♥❝❡ ❢✉♥❝t✐♦♥ ✇r✐t❡s ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡

❝♦♣✉❧❛ C✱

Q(X, X ) = 4 Z

C(u)dC(u) − 1, ✭✸✮

✐♥ ♣❛r❛❧❧❡❧ ✇✐t❤ t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✳

✹✳✸✳ ❛✳s✳ ❝♦♥s✐st❡♥❝② ♦❢ ❝♦♥❝♦r❞❛♥❝❡ ♠❡❛s✉r❡s ❢♦r ♣✉r❡❧② ❞✐s❝r❡t❡ ✈❡❝t♦rs

▲❡t X ∈ R d ❜❡ ❛ ♣✉r❡❧② ❞✐s❝r❡t❡✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❡❝t♦r ✇✐t❤ ❝❞❢ F ✱ X 1 , . . . , X n , . . .

❜❡ ❛♥ ❡r❣♦❞✐❝ s❛♠♣❧❡ ♦❢ X ❛♥❞ F n , G n t❤❡ ❡♠♣✐r✐❝❛❧ ❝❞❢s✳ ❈♦♥str✉❝t✱ ❛s ✐♥ s❡❝t✐♦♥ ✸✱

❝♦♥❞✐t✐♦♥❛❧❧② ♦♥ t❤❡ s❛♠♣❧❡✱ ❛ s❡q✉❡♥❝❡ X ˆ n ∼ F n ✱ X ∼ F ✱ t❤❡ ❝♦♣✉❧❛ r❡♣r❡s❡♥t❡rs U n = G n ( X n , V ) ∼ C n ✱ U = G ( X , V ) ∼ C ❛ss♦❝✐❛t❡❞ ✇✐t❤ X n ✱ X ✱ ✇✐t❤ V ♦❢

✐♥❞❡♣❡♥❞❡♥t ♠❛r❣✐♥❛❧s✱ ❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✳

◆♦t❡ t❤❛t ❡q✉❛t✐♦♥ ✭✸✮ ✇r✐t❡s✱ τ (F ) = 4E [C(U )] − 1✱ s♦ t❤❛t ✐t ❜❡❝♦♠❡s str❛✐❣❤t✲

❢♦r✇❛r❞ t♦ ❞❡✜♥❡ t❤❡ ❡♠♣✐r✐❝❛❧ ❝♦✉♥t❡r♣❛rt ♦❢ ❑❡♥❞❛❧❧✬s ❝♦❡✣❝✐❡♥t✿ ❞❡♥♦t❡

τ(F n ) := 4E [C n 1 (U n )] − 1,

✇❤❡r❡ ❡①♣❡❝t❛t✐♦♥ E ✐s ❝♦♥❞✐t✐♦♥❛❧ ♦♥ t❤❡ s❛♠♣❧❡✳ ❚❤❡ ❛❧♠♦st s✉r❡ ❝♦♥s✐st❡♥❝② ✇✳r✳t✳

t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ✐s ♣r♦✈❡❞✱ ❛s ❛ s✐♠♣❧❡ ❝♦r♦❧❧❛r② ♦❢ t❤❡♦r❡♠ ✸✳✶✱

❈♦r♦❧❧❛r② ✹✳✶✳ ❲✐t❤ P ✲♣r♦❜❛❜✐❧✐t② ♦♥❡✱

τ(F n ) a.s. → τ(F ),

❛s n → ∞ ✳

Pr♦♦❢✳ ❆s ✐♥ t❤❡ ♣r❡✈✐♦✉s ♣r♦♦❢s✱ ♦♥❡ ✇♦r❦s ❝♦♥❞✐t✐♦♥❛❧❧② ♦♥ t❤❡ s❛♠♣❧❡✱ ✐✳❡✳ ❢♦r ❛ ✜①❡❞

ω ∈ Ω 0 ✇✐t❤ P (ω 0 ) = 1✳ ❚❤❡♥✱

|C n 1 (U n ) − C(U )| ≤ ||C n 1 − C|| ∞ + |C(U n ) − C(U )|

❇♦t❤ t❡r♠s ♦♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ t❤❡ ✐♥❡q✉❛❧✐t② ❣♦❡s t♦ ③❡r♦ ❛✳s✳✱ t❤❡ ✜rst ♦♥❡ ❜②

❝♦r♦❧❧❛r② ✸✳✷ ❛♥❞ t❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ✐❞❡♥t✐❝❛❧❧② ③❡r♦ ❢♦r n ≥ N ✱ ❜② t❤❡♦r❡♠ ✸✳✶✳ ❯♥✐❢♦r♠

❜♦✉♥❞❡❞♥❡ss ♦❢ t❤❡s❡ t❡r♠s ②✐❡❧❞ t❤❡ ❞❡s✐r❡❞ r❡s✉❧t ❜② ❞♦♠✐♥❛t❡❞ ❝♦♥✈❡r❣❡♥❝❡✳ ❚❤✐s r❡❛s♦♥✐♥❣ ✐s ✈❛❧✐❞ ❢♦r ❡✈❡r② ✜①❡❞ ω ∈ Ω 0 ✇✐t❤ P (ω 0 ) = 1✱ ❤❡♥❝❡ t❤❡ ❛ss❡rt✐♦♥✳

❆♣♣❡♥❞✐①

Pr♦♦❢ ♦❢ t❤❡♦r❡♠ ✷✳✶✳ ▲❡t X 1 , X 2 , . . . , X ❜❡ ❞✐s❝r❡t❡ r❛♥❞♦♠ ✈❡❝t♦rs ✇✐t❤ ✈❛❧✉❡s ✐♥ ❛

✜♥✐t❡ ♦r ❝♦✉♥t❛❜❧❡ s❡t E✱ s✳t✳

P(X n = x) → P (X ∞ = x), ❛s n →, ∞, ∀x ∈ E. ✭✹✮

▲❡t q 0 := 0 ❛♥❞ q n (x) := inf n≤k<∞ P (X k = x)✳ ❚❤❡♥✱ ✭✹✮ ❡♥t❛✐❧s t❤❛t✱ ❢♦r ❛❧❧ x ∈ E✱

q n (x) ↑ P(X ∞ = x), ❛s n → ∞. ✭✺✮

✶✵

(12)

▲❡t N , V 1 , V 2 , . . . , W 1 , W 2 , . . . ❜❡ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ❡❧❡♠❡♥ts s✉❝❤ t❤❛t ❢♦r 1 ≤ n <

∞ ❛♥❞ x ∈ E✱

P(N = n) = X

x ∈E

q n (x) − X

x ∈E

q n−1 (x)

P (V n = x) =

( q

n

( x )−q

n−1

( x )

P(N=n) ✐❢ P (N = n) > 0

❛r❜✐tr❛r② ✐❢ P (N = n) = 0 P (W n = x) =

( P( X

n

= x )−q

n

( x )

P(N>n) ✐❢ P (N > n) > 0

❛r❜✐tr❛r② ✐❢ P (N > n) = 0.

❇② ❞♦♠✐♥❛t❡❞ ❝♦♥✈❡r❣❡♥❝❡✱ ✭✺✮ ❡♥t❛✐❧s P(N ≤ n) = X

x ∈E

q n ( x ) ↑ X

x ∈E

P( X = x ) = 1,

❛s n → ∞ ✱ ✐✳❡✳ N ✐s ✜♥✐t❡✳ ❉❡✜♥❡✱ ❢♦r 1 ≤ n ≤ ∞ ✱

X ˆ n =

( V N ✐❢ n ≥ N , W n ✐❢ n < N ,

❚❤❡♥ X ˆ ∞ = V N ❛♥❞ s✐♠♣❧❡ ❝♦♠♣✉t❛t✐♦♥s s❤♦✇ t❤❛t P( X ˆ n = x) = P(X n = x) ❛♥❞

P( X ˆ ∞ = x) = P (X ∞ = x) ❢♦r ❡❛❝❤ x ∈ E✳

❬✶❪ P❛✉❧ ❉❡❤❡✉✈❡❧s✳ ❈❛r❛❝tér✐s❛t✐♦♥ ❝♦♠♣❧èt❡ ❞❡s ❧♦✐s ❡①trê♠❡s ♠✉t✐✈❛r✐é❡s ❡t ❞❡ ❧❛ ❝♦♥✈❡r❣❡♥❝❡ ❞❡s t②♣❡s ❡①trê♠❡s✳ P✉❜✳ ■♥st✳ ❙t❛t✳ ❯♥✐✈✳ P❛r✐s✱ ✷✸✭✸✲✹✮✿✶✕✸✻✱ ✶✾✼✽✳

❬✷❪ P❛✉❧ ❉❡❤❡✉✈❡❧s✳ ▲❛ ❢♦♥❝t✐♦♥ ❞❡ ❞é♣❡♥❞❛♥❝❡ ❡♠♣✐r✐q✉❡ ❡t s❡s ♣r♦♣r✐étés✱ ✉♥ t❡st ♥♦♥ ♣❛r❛♠étr✐q✉❡

❞✬✐♥❞é♣❡♥❞❛♥❝❡✳ ❇✉❧❧❡t✐♥ ❞❡ ❧✬❆❝❛❞é♠✐❡ ❘♦②❛❧❡ ❞❡ ❇❡❧❣✐q✉❡✱ ✻✺✭❢✳✻✮✿✷✼✹✕✷✾✷✱ ✶✾✼✾✳

❬✸❪ P❛✉❧ ❉❡❤❡✉✈❡❧s✳ ❆ ♠✉❧t✐✈❛r✐❛t❡ ❇❛❤❛❞✉r✲❑✐❡❢❡r r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r t❤❡ ❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛ ♣r♦❝❡ss✳

❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡s✱ ✶✻✸✿✸✽✷✕✸✾✽✱ ◆♦✈❡♠❜❡r ✷✵✵✾✳

❬✹❪ ▼✐❝❤❡❧ ❉❡♥✉✐t ❛♥❞ P❤✐❧✐♣♣❡ ▲❛♠❜❡rt✳ ❈♦♥str❛✐♥ts ♦♥ ❝♦♥❝♦r❞❛♥❝❡ ♠❡❛s✉r❡s ✐♥ ❜✐✈❛r✐❛t❡ ❞✐s❝r❡t❡

❞❛t❛✳ ❏♦✉r♥❛❧ ♦❢ ▼✉❧t✐✈❛r✐❛t❡ ❆♥❛❧②s✐s✱ ✾✸✭✶✮✿✹✵ ✕ ✺✼✱ ✷✵✵✺✳

❬✺❪ ❘✳ ▼✳ ❉✉❞❧❡②✳ ❘❡❛❧ ❆♥❛❧②s✐s ❛♥❞ Pr♦❜❛❜✐❧✐t②✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✷♥❞ ❡❞✐t✐♦♥✱ ❆✉❣✉st

✷✵✵✷✳

❬✻❪ ❖❧✐✈✐❡r P✳ ❋❛✉❣❡r❛s✳ ❙❦❧❛r✬s t❤❡♦r❡♠ ❞❡r✐✈❡❞ ✉s✐♥❣ ♣r♦❜❛❜✐❧✐st✐❝ ❝♦♥t✐♥✉❛t✐♦♥ ❛♥❞ t✇♦ ❝♦♥s✐st❡♥❝② r❡s✉❧ts✳ ❏♦✉r♥❛❧ ♦❢ ▼✉❧t✐✈❛r✐❛t❡ ❆♥❛❧②s✐s✱ ✶✷✷✿✷✼✶ ✕ ✷✼✼✱ ✷✵✶✸✳

❬✼❪ ❏❡❛♥✲❉❛✈✐❞ ❋❡r♠❛♥✐❛♥✱ ❉r❛❣❛♥ ❘❛❞✉❧♦✈✐❝✱ ❛♥❞ ▼❛rt❡♥ ❲❡❣❦❛♠♣✳ ❲❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❡♠♣✐r✐❝❛❧

❝♦♣✉❧❛ ♣r♦❝❡ss❡s✳ ❇❡r♥♦✉❧❧✐✱ ✶✵✭✺✮✿✽✹✼✕✽✻✵✱ ❖❝t♦❜❡r ✷✵✵✹✳

❬✽❪ ❆❧❡①❛♥❞❡r ▼✳ ▲✐♥❞♥❡r ❛♥❞ ❆❧❡①❛♥❞❡r ❙③✐♠❛②❡r✳ ❆ ❧✐♠✐t t❤❡♦r❡♠ ❢♦r ❝♦♣✉❧❛s✳ Pr❡♣r✐♥t✱ ❉✐s❝✉ss✐♦♥

P❛♣❡r ✹✸✸✱ ❙♦♥❞❡r❢♦rs❝❤✉♥❣s❜❡r❡✐❝❤ ✸✽✻✱ ❆✈❛✐❧❛❜❧❡ ❛t ❡♣✉❜✳✉❜✳✉♥✐✲♠✉❡♥❝❤❡♥✳❞❡✱ ✷✵✵✺✳

❬✾❪ ❚♦r❣♥② ▲✐♥❞✈❛❧❧✳ ▲❡❝t✉r❡s ♦♥ t❤❡ ❝♦✉♣❧✐♥❣ ♠❡t❤♦❞✳ ❏✳ ❲✐❧❡② ❛♥❞ ❙♦♥s✱ ✶✾✾✷✳

❬✶✵❪ ▼✳ ▼❡s✜♦✉✐ ❛♥❞ ❏✳✲❋✳ ◗✉❡ss②✳ ❈♦♥❝♦r❞❛♥❝❡ ♠❡❛s✉r❡s ❢♦r ♠✉❧t✐✈❛r✐❛t❡ ♥♦♥✲❝♦♥t✐♥✉♦✉s r❛♥❞♦♠

✈❡❝t♦rs✳ ❏♦✉r♥❛❧ ♦❢ ▼✉❧t✐✈❛r✐❛t❡ ❆♥❛❧②s✐s✱ ✶✵✶✿✷✸✾✽✕✷✹✶✵✱ ✷✵✶✵✳

❬✶✶❪ ❉❛✈✐❞ ❙✳ ▼♦♦r❡ ❛♥❞ ▼✳ ❈✳ ❙♣r✉✐❧❧✳ ❯♥✐✜❡❞ ❧❛r❣❡✲s❛♠♣❧❡ t❤❡♦r② ♦❢ ❣❡♥❡r❛❧ ❝❤✐✲sq✉❛r❡❞ st❛t✐st✐❝s ❢♦r t❡sts ♦❢ ✜t✳ ❚❤❡ ❆♥♥❛❧s ♦❢ ❙t❛t✐st✐❝s✱ ✸✭✸✮✿✺✾✾✕✻✶✻✱ ▼❛② ✶✾✼✺✳

❬✶✷❪ ❘♦❣❡r ❇✳ ◆❡❧s❡♥✳ ❆♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ ❝♦♣✉❧❛s✳ ❙♣r✐♥❣❡r✱ ✷✵✵✻✳

❬✶✸❪ ❏✳ ◆❡s❧❡❤♦✈❛✳ ❖♥ r❛♥❦ ❝♦rr❡❧❛t✐♦♥ ♠❡❛s✉r❡s ❢♦r ♥♦♥✲❝♦♥t✐♥✉♦✉s r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ❏♦✉r♥❛❧ ♦❢

▼✉❧t✐✈❛r✐❛t❡ ❆♥❛❧②s✐s✱ ✾✽✿✺✹✹✕✺✻✼✱ ✷✵✵✼✳

❬✶✹❪ ▲✉❞❣❡r ❘üs❝❤❡♥❞♦r❢✳ ❆s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥s ♦❢ ♠✉❧t✐✈❛r✐❛t❡ r❛♥❦ ♦r❞❡r st❛t✐st✐❝s✳ ❚❤❡ ❆♥♥❛❧s ♦❢

❙t❛t✐st✐❝s✱ ✹✭✺✮✿✾✶✷✕✾✷✸✱ ❙❡♣t❡♠❜❡r ✶✾✼✻✳

✶✶

(13)

❬✶✺❪ ▲✉❞❣❡r ❘üs❝❤❡♥❞♦r❢✳ ❙t♦❝❤❛st✐❝❛❧❧② ♦r❞❡r❡❞ ❞✐str✐❜✉t✐♦♥s ❛♥❞ ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ ♦❝✲❢✉♥❝t✐♦♥ ♦❢

s❡q✉❡♥t✐❛❧ ♣r♦❜❛❜✐❧✐t② r❛t✐♦ t❡sts✳ ▼❛t❤✳ ❖♣❡r❛t✐♦♥❢♦rs❝❤✳ ❙t❛t✐st✳✱ ❙❡r✐❡s ❙t❛t✐st✐❝s✱ ✶✷✭✸✮✿✸✷✼✕✸✸✽✱

✶✾✽✶✳

❬✶✻❪ ▲✉❞❣❡r ❘üs❝❤❡♥❞♦r❢✳ ❖♥ t❤❡ ❞✐str✐❜✉t✐♦♥❛❧ tr❛♥s❢♦r♠✱ s❦❧❛r✬s t❤❡♦r❡♠✱ ❛♥❞ t❤❡ ❡♠♣✐r✐❝❛❧ ❝♦♣✉❧❛

♣r♦❝❡ss✳ ❏♦✉r♥❛❧ ♦❢ ❙t❛t✐st✐❝❛❧ P❧❛♥♥✐♥❣ ❛♥❞ ■♥❢❡r❡♥❝❡✱ ✶✸✾✭✶✶✮✿✸✾✷✶✕✸✾✷✼✱ ✷✵✵✾✳

❬✶✼❪ ▲✉❞❣❡r ❘üs❝❤❡♥❞♦r❢✳ ▼❛t❤❡♠❛t✐❝❛❧ ❘✐s❦ ❆♥❛❧②s✐s ✲ ❉❡♣❡♥❞❡♥❝❡✱ ❘✐s❦ ❇♦✉♥❞s✱ ❖♣t✐♠❛❧ ❆❧❧♦❝❛t✐♦♥s

❛♥❞ P♦rt❢♦❧✐♦s✳ ❙♣r✐♥❣❡r✱ ✷✵✶✸✳

❬✶✽❪ ❆❜❡ ❙❦❧❛r✳ ❋♦♥❝t✐♦♥s ❞❡ ré♣❛rt✐t✐♦♥ à ♥ ❞✐♠❡♥s✐♦♥s ❡t ❧❡✉rs ♠❛r❣❡s✳ P✉❜❧✐❝❛t✐♦♥s ❞❡ ❧✬■♥st✐t✉t ❞❡

❙t❛t✐st✐q✉❡ ❞❡ ❧✬❯♥✐✈❡rs✐té ❞❡ P❛r✐s✱ ✽✿✷✷✾✕✷✸✶✱ ✶✾✺✾✳

❬✶✾❪ ❆❜❡ ❙❦❧❛r✳ ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥s✱ ❛♥❞ ❝♦♣✉❧❛s✳ ❑②❜❡r♥❡t✐❦❛✱ ✾✭✻✮✿✹✹✾✕✹✻✵✱

✶✾✼✸✳

❬✷✵❪ ❆✳ ❱✳ ❙❦♦r♦❦❤♦❞✳ ▲✐♠✐t t❤❡♦r❡♠s ❢♦r st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✳ ❚❤❡♦r② ♦❢ Pr♦❜❛❜✐❧✐t② ❛♥❞ ✐ts ❆♣♣❧✐❝❛✲

t✐♦♥s✱ ✶✿✷✻✶✕✷✾✵✱ ✶✾✺✻✳

❬✷✶❪ ❍❡r♠❛♥♥ ❚❤♦r✐ss♦♥✳ ❈♦✉♣❧✐♥❣✱ st❛t✐♦♥❛r✐t②✱ ❛♥❞ r❡❣❡♥❡r❛t✐♦♥✳ ❙♣r✐♥❣❡r✱ ✷✵✵✵✳

✶✷

Références

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