trt t q - - PowerPoint PPT Presentation

❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥

▲➪❙❩▲Ó ●❊❘❊◆❈❙➱❘✶✱

✇✐t❤ ❈✳ Pr♦s❞♦❝✐♠✐✷ ❛♥❞ ❩s✳ ❱á❣ó✸

✶▼❚❆ ❙❩❚❆❑■✱ ✷▲❯■❙❙ ❯♥✐✈❡rs✐t②✱ ✸PP❑❊ ■❚❑

❚❤❡ ◆✐♥t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ▼❛tr✐①✲❆♥❛❧②t✐❝ ▼❡t❤♦❞s ✐♥ ❙t♦❝❤❛st✐❝ ▼♦❞❡❧s

❇✉❞❛♣❡st✱ ❏✉♥❡ ✷✽ ✲ ✸✵✱ ✷✵✶✻

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶ ✴

❚❍❊ ❉❨◆❆▼■❈❙ ♦❢ ❛ ◗❯❊❯❊

❈♦♥s✐❞❡r ❛ s✐♥❣❧❡ s❡r✈❡r q✉❡✉❡✳ ❲❛✐t✐♥❣ t✐♠❡ ♦❢ t❤❡ n✲t❤ ❝✉st♦♠❡r✿ Wn. ❚❤❡ ❞②♥❛♠✐❝s ♦❢ Wn ✐s ❣✐✈❡♥ ❜② ❛ ♥♦♥✲❧✐♥❡❛r s②st❡♠✿ Wn = (Wn−✶ + Xn)+ ✇✐t❤ W✵ = ✵, ✭✶✮ ✇❤❡r❡ Xn = Vn−✶ − Un = s❡r✈✐❝❡ t✐♠❡ ♠✐♥✉s ✐♥t❡r❛rr✐✈❛❧ t✐♠❡✳ ❆ s②st❡♠✲t❤❡♦r❡t✐❝ ♣♦✐♥t ♦❢ ✈✐❡✇✿ ✭✽✮ ✐s ♥♦t ❛ st❛❜❧❡ s②st❡♠✳ ❆ s✐♠✐❧❛r ♥♦♥✲❧✐♥❡❛r ❞②♥❛♠✐❝s ❛r✐s❡s ✐♥ t❤❡ t❤❡♦r② ♦❢ r✐s❦ ♣r♦❝❡ss❡s✿ W −

n = (W − n−✶ + X − n )−

✇✐t❤ W −

✵ = K > ✵.

✭✷✮ ✳

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✷ ✴

P❘❖❇❆❇■▲■❙❚■❈ ❙❚❆❇■▲■❚❨ ♦❢ ❛ ◗❯❊❯❊

❆ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥✿ ❛ss✉♠❡ ✐✳✐✳❞✳ ✐♥♣✉ts (Xn)✱ ✇✐t❤ ❊(Xn) < ✵✳ ▼❛r❦♦✈✐❛♥ t❡❝❤♥✐q✉❡s✿ ❡st❛❜❧✐s❤ ❣❡♦♠❡tr✐❝ ❡r❣♦❞✐❝✐t② ❛ss✉♠✐♥❣ E(❡①♣ c′X✶) < ✶ ❢♦r s♦♠❡ c′ > ✵. ✭✸✮ ❙tr♦♥❣ ▲▲◆ ❢♦❧❧♦✇s ❢♦r ❢✉♥❝t✐♦♥s ♦❢ Wn. ❙❡❡ ▼❡②♥ ✫ ❚✇❡❡❞✐❡✳

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✸ ✴

P❘❖❇▲❊▼ ❙❚❆❚❊▼❊◆❚

❯♥❞❡r ✇❤❛t ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ✐♥♣✉ts (Xn) ❝❛♥ ✇❡ ❡♥s✉r❡✿ ✶✳ ❆ str♦♥❣ ▲▲◆ ❢♦r t❤❡ ❡♠♣✐r✐❝❛❧ t❛✐❧ ♣r♦❜❛❜✐❧✐t✐❡s✿ ❧✐♠ s✉♣

N

✶ N

N

• n=✶

I{Wn>K}≤ ❧✐♠ s✉♣

n

P(Wn > K) ❛✳s✳ ✷✳ ❊①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ♦❢ t❛✐❧ ♣r♦❜❛❜✐❧✐t✐❡s✿ P(Wn > K) < Ce−cK. ❚❡❝❤♥✐❝❛❧ ❦✐♥s❤✐♣ ♦❢ t❤❡ t✇♦ ♣r♦❜❧❡♠s✳

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✹ ✴

❈❍❆◆●❊ ❉❊❚❊❈❚■❖◆

❚❤❡ ♣r♦❜❧❡♠✿ ❞❡t❡❝t ❝❤❛♥❣❡s ♦❢ st❛t✐st✐❝❛❧ ♣❛tt❡r♥s ♦❢ s✐❣♥❛❧s ✐♥ r❡❛❧ t✐♠❡✳ ❊①❛♠♣❧❡✿ ♠♦♥✐t♦r✐♥❣ ❊❊● s✐❣♥❛❧s ❢♦r ❡♣✐❧❡♣t✐❝ ♣❛t✐❡♥ts ❙❡❡✿ ❱✳P♦♦r ❛♥❞ ❖✳❍❛❞❥✐❧✐❛❞✐s ✭✷✵✵✾✮✿ ◗✉✐❝❦❡st ❉❡t❡❝t✐♦♥✳

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✺ ✴

❆ ❈▲❆❙❙■❈ P❘❖❇▲❊▼

• ✐✈❡♥ ❛ s❡q✉❡♥❝❡ ♦❢ ✐✳✐✳❞✳ r✳✈✳✲s Yn ✇✐t❤ ♣r♦❜✳ ❞❡♥s✐t② ❢✉♥❝t✐♦♥s

f (y, θ✵) ❢♦r n < τ ❛♥❞ f (y, θ✶) ❢♦r n ≥ τ. ❊st✐♠❛t❡ ❝❤❛♥❣❡ ♣♦✐♥t ✿ τ ✉s✐♥❣ ♦❜s❡r✈❛t✐♦♥s (yn)✳ ❚❤❡ ❈✉♠✉❧❛t✐✈❡ ❙✉♠ ✭❈❯❙❯▼✮ t❡st ♦r P❛❣❡✲❍✐♥❦❧❡② ❞❡t❡❝t♦r✿ ❊✳❙✳ P❛❣❡✱ ❇✐♦♠❡tr✐❦❛✱ ✶✾✺✹ ❉✳❱✳ ❍✐♥❦❧❡②✱ ❏✳❆♠❡r✳ ❙t❛t✐st✳ ❆ss♦❝✳✱ ✶✾✼✶

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✻ ✴

❙❚❆❚■❙❚■❈❙ ❛♥❞ ■❚

❆ ♠♦❞❡r♥ ✐♥t❡r♣r❡t❛t✐♦♥✱ ❢♦❧❧♦✇✐♥❣ ❏✳❘✐ss❛♥❡♥✱ ✶✾✽✾✿ ❊♥❝♦❞❡ ❞❛t❛ ✉s✐♥❣ t❤❡ t✇♦ ♣♦ss✐❜❧❡ ♠♦❞❡❧s✱ ❢♦❧❧♦✇✐♥❣ ■♥❢✳❚❤②✳✿ ❚❤❡ q✉❛s✐✲♦♣t✐♠❛❧ ❝♦❞❡✲❧❡♥❣t❤s ❛r❡ − ❧♦❣ f (yn, θ✵) ❛♥❞ − ❧♦❣ f (yn, θ✶). ❚❤❡ ❞✐✛❡r❡♥❝❡s ✐♥ ❝♦❞❡✲❧❡♥❣t❤s ❞❡✜♥❡ t❤❡ s❝♦r❡ Xn = − ❧♦❣ f (yn, θ✵) + ❧♦❣ f (yn, θ✶). ◆♦✇ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ✐♥❡q✉❛❧✐t② ❣✐✈❡s ❊Xn < ✵ ❢♦r n < τ ❛♥❞ ❊Xn > ✵ ❢♦r n ≥ τ.

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✼ ✴

❚❍❊ ❈❯❙❯▼ ❚❊❙❚ ❢♦r ■✳■✳❉✳ ❉❆❚❆

▲❡t S✵ = ✵ ❛♥❞ ❧❡t Sn =

n

• k=✶

Xk ❢♦r n ≥ ✶. ❚❤❡♥ ❊Sn ❤❛s ❛ ♠✐♥✐♠✉♠ ❛t τ − ✶. ❚❛s❦✿ ❛♣♣r♦①✐♠❛t❡ ♦♥✲❧✐♥❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ Sn. ❚❤❡ ❈❯❙❯▼ st❛t✐st✐❝s ♦r P❛❣❡✲❍✐♥❦❧❡② ❞❡t❡❝t♦r✿ ❞❡✜♥❡ gn = Sn − ♠✐♥

✵≤k≤n Sk.

• ❡♥❡r❛t❡ ❛♥ ❛❧❛r♠ ✐❢ gn > δ✱ ✇✐t❤ s♦♠❡ ✜①❡❞ t❤r❡s❤♦❧❞ δ > ✵.

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✽ ✴

❋❆▲❙❊ ❆▲❆❘▼ ❘❆❚❊

❆♣♣❧② t❤❡ P❛❣❡✲❍✐♥❦❧❡② ❞❡t❡❝t♦r t♦ ❛ ♣r♦❝❡ss ✇✐t❤ ♥♦ ❝❤❛♥❣❡ ❛t ❛❧❧✳ ❆ ❦❡② ♣❡r❢♦r♠❛♥❝❡ ❝❤❛r❛❝t❡r✐st✐❝s✿ ❢❛❧s❡ ❛❧❛r♠ ♣r♦❜❛❜✐❧✐t② ❧✐♠ s✉♣

n

Pθ✵(gn > δ). Pr❛❝t✐❝❛❧ r❡❧❡✈❛♥❝❡✿ ❢❛❧s❡ ❛❧❛r♠ r❛t❡ ✭❋❆❘✮ ❞❡✜♥❡❞ ❛s ❛✳s✳ ❧✐♠ s✉♣

N

✶ N

N

• n=✶

I{gn>δ}. Pr♦❜❧❡♠✿ ✜♥❞ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ❢♦r t❤❡ ❋❆❘✳

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✾ ✴

❚❍❊ ❉❨◆❆▼■❈❙ ♦❢ ❈❯❙❯▼

❚❤❡ ❞②♥❛♠✐❝s ♦❢ gn ✐s ❡❛s✐❧② ♦❜t❛✐♥❡❞ ❛s ❢♦❧❧♦✇s✿ gn = (gn−✶ + Xn)+ ✇✐t❤ g✵ = ✵. ❚❤✐s ❡st❛❜❧✐s❤❡s t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ q✉❡✉✐♥❣✱ Wn✱ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥✱ gn✳ ❖❜❥❡❝t✐✈❡✿ ❜♦✉♥❞✐♥❣ t❤❡ ❡♠♣✐r✐❝❛❧ t❛✐❧ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ gn. ❚✇♦ r❡❧❛t❡❞ t❡❝❤♥✐❝❛❧ ♣r♦❜❧❡♠s✿ ❊①♣♦♥❡♥t✐❛❧ ❜♦✉♥❞s ❢♦r t❤❡ t❛✐❧ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ gn. ▼✐①✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ gn.

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶✵ ✴

▼❖❚■❱❆❚■❖◆ ❢♦r ▼■❳■◆●

▲❡t (νn) ❜❡ ❛♥ Rs✲✈❛❧✉❡❞ ✐✳✐✳❞✳ s❡q✉❡♥❝❡ ♦❢ r✳✈✳✲s s✉❝❤ t❤❛t s✉♣

n≥✵

E |νn|q < +∞ ❢♦r ❛❧❧ ✶ ≤ q < ∞. ▲❡t t❤❡ s × s ♠❛tr✐① A ❜❡ st❛❜❧❡✱ ❛♥❞ ❞❡✜♥❡ t❤❡ ✜❧t❡r❡❞ ♣r♦❝❡ss Xn = AXn−✶ + νn ✇✐t❤ X✵ = ✵. ❉❡❝♦♠♣♦s❡ (Xn) ❛s Xn = AτXn−τ +

τ−✶

• k=✵

Akνn−k.

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶✶ ✴

L✲▼■❳■◆●✱ ■✳

❉❡✜♥✐t✐♦♥

▲❡t X = (Xn) ❜❡ ❛ st♦❝❤❛st✐❝ ♣r♦❝❡ss ♦♥ (Ω, F, P)✳ X ✐s M✲❜♦✉♥❞❡❞ ✐❢ ❢♦r ❛❧❧ ✶ ≤ q < +∞ Mq(X) := s✉♣

n≥✵

Xn q< +∞. ▲❡t ν = (νn) ❜❡ ❛♥ ✐✳✐✳❞✳ s❡q✉❡♥❝❡✱ ❛♥❞ ❞❡✜♥❡ ✐ts ♣❛st ❛♥❞ ❢✉t✉r❡ ❛s Fn = σ(νk : k ≤ n) ❛♥❞ F+

n = σ(νk : k ≥ n + ✶).

▲❡t τ > ✵ ❜❡ ❛♥ ✐♥t❡❣❡r✱ ❛ ✜①❡❞ ♠❡♠♦r② ❧❡♥❣t❤✱ ❛♥❞ ❞❡✜♥❡❞ ❢♦r ✶ ≤ q < +∞ t❤❡ ❡rr♦r ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❜② t❤❡ ♥❡❛r ♣❛st ❛s γq(τ, X) = γq(τ) := s✉♣

n≥τ

Xn − ❊(Xn|F+

n−τ) q .

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶✷ ✴

L✲▼■❳■◆●✱ ■■✳

❉❡✜♥✐t✐♦♥

❆ st♦❝❤❛st✐❝ ♣r♦❝❡ss X = (Xn) ✐s L✲♠✐①✐♥❣ ✇✳r✳t✳ (Fn, F+

n ) ✐❢ ✐t ✐s ❛❞❛♣t❡❞

✐✳❡✳ Xn ✐s Fn✲♠❡❛s✉r❛❜❧❡ ❢♦r ❛❧❧ n ≥ ✶✱ X ✐s M✲❜♦✉♥❞❡❞✱ ❛♥❞ Γq(X) :=

+∞

• τ=✵

γq(τ) < +∞ ❢♦r ❛❧❧ ✶ ≤ q < +∞. ❘❡♠❛r❦✿ ✇❡ ❝❛♥ ❛❧s♦ ❤❛✈❡ q = ∞✳ ❘❡❢❡r❡♥❝❡s✿ ▲❥✉♥❣✱ ▲✳✿ ▼❛t❤✳ Pr♦❣r❛♠♠✐♥❣✱ ✶✾✼✻✳ ▲●✿ ❙t♦❝❤❛st✐❝s✱ ✶✾✽✾✳

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶✸ ✴

❚❍▼✿ ❢♦r ■✳■✳❉✳ ❙❈❖❘❊❙ gn ✐s L✲▼■❳■◆●

❆ss✉♠❡ E(X✶) < ✵, ❛♥❞ ❛❧s♦ E(❡①♣ cX✶) < ∞ ❢♦r s♦♠❡ c > ✵✳ ❚❤❡♥ µ := E(❡①♣ c′X✶) < ✶ ❢♦r s♦♠❡ c′ > ✵. ✭✹✮ ▲❡t Fn := σ(Xi | i ≤ n) ❛♥❞ F+

n := σ(Xi | i ≥ n + ✶).

❚❤❡♦r❡♠

▲❡t (Xn) ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ✐✳✐✳❞✳ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t ✭✹✮ ❤♦❧❞s✳ ❚❤❡♥ t❤❡ P❛❣❡✲❍✐♥❦❧❡② ❞❡t❡❝t♦r (gn) ✐s L✲♠✐①✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ (Fn, F+

n )✳

▲● ❛♥❞ Pr♦s❞♦❝✐♠✐✱ ❈✳ ❙②st❡♠s ✫ ❈♦♥tr♦❧ ▲❡tt❡rs✱ ✷✵✶✶✳

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶✹ ✴

❊▼P■❘■❈❆▲ ❚❆■▲ P❘❖❇❆❇■▲■❚■❊❙

❆ ❦♥♦✇♥ r❡s✉❧t ✐♥ r✐s❦ t❤❡♦r②✿ ❢♦r ❛♥② c′′ s✉❝❤ t❤❛t ✵ < c′′ < c′✱ ✇❡ ❤❛✈❡ s✉♣

n

E

• ❡①♣ c′′ gn
• < ∞.

✭✺✮ ❍❡♥❝❡ P(gn > δ) ≤ Ce−c′′δ ✇✐t❤ s♦♠❡ ✵ < C < ∞. ❊q✉✐✈❛❧❡♥t❧②✱ E I{gn>δ} ≤ Ce−c′′δ. ❙✐♥❝❡ (gn) ✐s L✲♠✐①✐♥❣✱ ❜② ❛ str♦♥❣ ▲▲◆ ✐t ❢♦❧❧♦✇s t❤❛t ❧✐♠ s✉♣

N

✶ N

N

• n=✶

I{gn>δ} ≤ C ′e−c′′δ.

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶✺ ✴

❉❊P❊◆❉❊◆❚ ❙❈❖❘❊❙

❚❤❡ ❦❡② t❡❝❤♥✐❝❛❧ ❝♦♥❞✐t✐♦♥ t♦ ❜❡ ❡st❛❜❧✐s❤❡❞ ❛❜♦✈❡✿ ❢♦r ❛♥② c′ > ✵ E ❡①♣

n

• k=✶

c′Xk ≤ Ce−c′′n ✇✐t❤ s♦♠❡ C, c′′ > ✵✳ ❲r✐t✐♥❣ t❤❡ ❧✳❤✳s✳ ❛s E ❡①♣ n

• k=✶

c′(Xk − EXk)

• · ❡①♣

n

• k=✶

c′ EXk

• ,

❛ss✉♠✐♥❣ EXk ≤ −ε✱ ✐t ✐s s✉✣❝✐❡♥t t♦ s❤♦✇ t❤❛t ❢♦r s♠❛❧❧ c′✲s ❛♥❞ ❛❧❧ n✿ E ❡①♣ n

• k=✶

c′(Xk − EXk)

• ≤ eκ(c′)✷n,

✇✐t❤ s♦♠❡ κ > ✵.

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶✻ ✴

❆◆ ❊❳P❖◆❊◆❚■❆▲ ▼❖▼❊◆❚ ❈❖◆❉■❚■❖◆

▲❡t X = (Xn) ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ r❡❛❧✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ❈♦♥❞✐t✐♦♥ ❊✳❚❤❡r❡ ❡①✐st c > ✵ ❛♥❞ κ > ✵ s✉❝❤ t❤❛t ❢♦r ✵ ≤ c′ < c ❛♥❞ ❛❧❧ ✶ ≤ m ≤ n ✇❡ ❤❛✈❡ ❊ ❡①♣

• c′

n

• k=m

(Xk − EXk)

• ≤ ❡①♣
• κ (c′)✷(n − m + ✶)
• .

❖❜❥❡❝t✐✈❡✿ ✜♥❞ ✉s❡❢✉❧ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❛❜♦✈❡ ✐♥❡q✉❛❧✐t② t♦ ❤♦❧❞✳

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶✼ ✴

■✳■✳❉✳ ❘❊❱■❙■❚❊❉

▲❡♠♠❛

▲❡t (Xn) ❜❡ ❛ ③❡r♦✲♠❡❛♥✱ ✐✳✐✳❞✳ s❡q✉❡♥❝❡ s✉❝❤ t❤❛t E ec|Xn| < ∞. ❚❤❡♥ ❈♦♥❞✐t✐♦♥ ❊ ✐s s❛t✐s✜❡❞✳ ❚❤❡ ♣r♦♦❢ ✐s tr✐✈✐❛❧✱ ♥♦t✐♥❣✿ Eec′X ≤ eκ(c′)✷. ❢♦r |c′| ≤ c ✇✐t❤ s♦♠❡ κ > ✵. ❲❡ ❣❡t ❡✈❡♥ ♠♦r❡✳

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶✽ ✴

❆◆❖❚❍❊❘ ❊❳P❖◆❊◆❚■❆▲ ▼❖▼❊◆❚ ■◆❊◗❯❆▲■❚❨

▲❡t X ❜❡ ❛ t✇♦✲s✐❞❡❞ ✐✳✐✳❞✳ s❡q✉❡♥❝❡ ❛s ❛❜♦✈❡✳ ▲❡t h = (hk), k = ✵, ✶, ... ❜❡ ❛♥ l✶ ✲ s❡q✉❡♥❝❡ ❛♥❞ ❞❡✜♥❡ Yn =

∞

• k=✵

hkXn−k ✐♥ s❤♦rt Y = h ⋆ X. ❲r✐t❡ h ✷

✷

= ∞

k=✵ h✷ k.

❚❤❡♦r❡♠

▲❡t Y = (Yn) ❜❡ ❛ ❛s ❛❜♦✈❡✳ ❚❤❡♥ ❢♦r h ✷≤ c E ❡①♣ (h ⋆ X) ≤ ❡①♣

• κ h ✷

✷

• .

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶✾ ✴

❆ ❙❊❈❖◆❉ ❊❳P❖◆❊◆❚■❆▲ ▼❖▼❊◆❚ ❈❖◆❉■❚■❖◆

▲❡t X = (Xn) ❜❡ ❛ t✇♦✲s✐❞❡❞ s❡q✉❡♥❝❡ ♦❢ r❡❛❧✲✈❛❧✉❡❞ r✳✈✳✲s✱ EXn = ✵. ❈♦♥❞✐t✐♦♥ ❙❊✳ ❚❤❡r❡ ❡①✐st c > ✵ ❛♥❞ κ > ✵ s✉❝❤ t❤❛t ❢♦r ❛♥② h ∈ l✶ ✇✐t❤ h ✷≤ c ✇❡ ❤❛✈❡ E ❡①♣ ∞

• k=✵

hkXn−k

• ≤ ❡①♣
• κ h ✷

✷

• .

✭✻✮ ◆♦t❡✿ ✐❢ X = (Xn) s❛t✐s✜❡s ❈♦♥❞✐t✐♦♥ ❙❊ ❛♥❞ g ∈ l✶ t❤❡♥ Y = g ⋆ X ❛❧s♦ s❛t✐s✜❡s ❈♦♥❞✐t✐♦♥ ❙❊✳ ✭❚r✐✈✐❛❧✮✳ ❊①❛♠♣❧❡✿ |Xn| ≤ K.

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✷✵ ✴

❆ ◆❖◆✲❈❖◆❙❚❘❯❈❚■❱❊ ❊❳❆▼P▲❊

❚❤❡♦r❡♠

▲❡t (Xn) ❜❡ ❛ ③❡r♦✲♠❡❛♥ L✲♠✐①✐♥❣ ♣r♦❝❡ss s✉❝❤ t❤❛t ✇❡ ❤❛✈❡ M∞(X) < +∞ ❛♥❞ Γ∞(X) < +∞. ✭✼✮ ❚❤❡♥ ❢♦r ❛♥② ❞❡t❡r♠✐♥✐st✐❝ s❡q✉❡♥❝❡ fn E ❡①♣ n

• k=✶

fnXn − ✷M∞(X)Γ∞(X)

n

• k=✶

f ✷

n

• ≤ ✶.

■t ❢♦❧❧♦✇s t❤❛t ❈♦♥❞✐t✐♦♥ ❙❊ ✐s s❛t✐s✜❡❞✳ ▲●✿ ❙t♦❝❤❛st✐❝s ❛♥❞ ❙t♦❝❤❛st✐❝ ❘❡♣♦rts✱ ✶✾✾✶✳

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✷✶ ✴

❆◆ ❖P❊◆ P❘❖❇▲❊▼✿ ❙❊❘■❆▲ ■◆❚❊❘❈❖◆◆❊❈❚■❖◆

▲❡t X = (Xn) ❜❡ ❛ t✇♦✲s✐❞❡❞ s❡q✉❡♥❝❡ ♦❢ r❡❛❧✲✈❛❧✉❡❞ r✳✈✳✲s✱ s❛t✐s❢②✐♥❣ EXn ≤ −ǫ < ✵ ❛♥❞ ✐♥ ❛❞❞✐t✐♦♥ ❈♦♥❞✐t✐♦♥ ❙❊✳ ❈♦♥s✐❞❡r ❛ s✐♥❣❧❡ s❡r✈❡r q✉❡✉❡ ✇✐t❤ ✇❛✐t✐♥❣ t✐♠❡ ❞❡♥♦t❡❞ ❜② Wn. ▲❡t t❤❡ ❞②♥❛♠✐❝s ♦❢ Wn ✐s ❣✐✈❡♥ ❜② t❤❡ ♥♦♥✲❧✐♥❡❛r s②st❡♠✿ Wn = (Wn−✶ + Xn)+ ✇✐t❤ W✵ = ✵. ✭✽✮ ❯♥❞❡r ✇❤❛t ❝♦♥❞✐t✐♦♥s ❞♦❡s (Wn) s❛t✐s❢② ❈♦♥❞✐t✐♦♥ ❙❊ ❄

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✷✷ ✴

❘❊❈❘❊❆❚■❖◆❆▲ ▼❆❚❍

❊st❛❜❧✐s❤ st❛❜✐❧✐t② ♣r♦♣❡rt✐❡s ♦❢ t❤❡ P❛❣❡✲❍✐♥❦❧❡②✲❞❡t❡❝t♦r ❢♦r ❞❡t❡r♠✐♥✐st✐❝ ✐♥♣✉ts✿ ❛ss✉♠❡ t❤❛t (Xn) ✐s ❛ ❞❡t❡r♠✐♥✐st✐❝ s❡q✉❡♥❝❡ s❛t✐s❢②✐♥❣ ❧✐♠ s✉♣

N− →+∞

✶ N

N

• n=✶

Xn < ✵. ✭✾✮ ▲❡t (gn) ❜❡ t❤❡ r❡s♣♦♥s❡ ♦❢ t❤❡ P❛❣❡✲❍✐♥❦❧❡②✲❞❡t❡❝t♦r ❞r✐✈❡♥ ❜② (Xn)✳ ❉♦❡s ✐t ❢♦❧❧♦✇ t❤❛t ❧✐♠ s✉♣

N→∞

✶ N

N

• n=✶

gn < ∞? ✭✶✵✮

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✷✸ ✴

❚❍❆◆❑ ❨❖❯ ❢♦r ❨❖❯❘ ❆❚❚❊◆❚■❖◆ ✦

▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✷✹ ✴