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❆♥ ✐♥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✿ W n . ❚❤❡ ❞②♥❛♠✐❝s ♦❢ W n ✐s ❣✐✈❡♥ ❜② ❛ ♥♦♥✲❧✐♥❡❛r s②st❡♠ ✿ ✇✐t❤ W ✵ = ✵ , ✭✶✮ W n = ( W n − ✶ + X n ) + ✇❤❡r❡ X n = V n − ✶ − U n = 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 − W − ✇✐t❤ ✵ = K > ✵ . ✭✷✮ n ) − ✳ ▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✷ ✴

P❘❖❇❆❇■▲■❙❚■❈ ❙❚❆❇■▲■❚❨ ♦❢ ❛ ◗❯❊❯❊ ❆ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥✿ ❛ss✉♠❡ ✐✳✐✳❞✳ ✐♥♣✉ts ( X n ) ✱ ✇✐t❤ ❊ ( X n ) < ✵✳ ▼❛r❦♦✈✐❛♥ t❡❝❤♥✐q✉❡s✿ ❡st❛❜❧✐s❤ ❣❡♦♠❡tr✐❝ ❡r❣♦❞✐❝✐t② ❛ss✉♠✐♥❣ c ′ > ✵ . E ( ❡①♣ c ′ X ✶ ) < ✶ ✭✸✮ ❢♦r s♦♠❡ ❙tr♦♥❣ ▲▲◆ ❢♦❧❧♦✇s ❢♦r ❢✉♥❝t✐♦♥s ♦❢ W n . ❙❡❡ ▼❡②♥ ✫ ❚✇❡❡❞✐❡✳ ▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✸ ✴

P❘❖❇▲❊▼ ❙❚❆❚❊▼❊◆❚ ❯♥❞❡r ✇❤❛t ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ✐♥♣✉ts ( X n ) ❝❛♥ ✇❡ ❡♥s✉r❡✿ ✶✳ ❆ str♦♥❣ ▲▲◆ ❢♦r t❤❡ ❡♠♣✐r✐❝❛❧ t❛✐❧ ♣r♦❜❛❜✐❧✐t✐❡s✿ N ✶ � ❧✐♠ s✉♣ I { W n > K } ≤ ❧✐♠ s✉♣ ❛✳s✳ P ( W n > K ) N n N n = ✶ ✷✳ ❊①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ♦❢ t❛✐❧ ♣r♦❜❛❜✐❧✐t✐❡s✿ P ( W n > 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 Y n ✇✐t❤ ♣r♦❜✳ ❞❡♥s✐t② ❢✉♥❝t✐♦♥s f ( y , θ ✵ ) n < τ f ( y , θ ✶ ) n ≥ τ. ❢♦r ❛♥❞ ❢♦r ❊st✐♠❛t❡ ❝❤❛♥❣❡ ♣♦✐♥t ✿ τ ✉s✐♥❣ ♦❜s❡r✈❛t✐♦♥s ( y n ) ✳ ❚❤❡ ❈✉♠✉❧❛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 ( y n , θ ✵ ) − ❧♦❣ f ( y n , θ ✶ ) . ❛♥❞ ❚❤❡ ❞✐✛❡r❡♥❝❡s ✐♥ ❝♦❞❡✲❧❡♥❣t❤s ❞❡✜♥❡ t❤❡ s❝♦r❡ X n = − ❧♦❣ f ( y n , θ ✵ ) + ❧♦❣ f ( y n , θ ✶ ) . ◆♦✇ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ✐♥❡q✉❛❧✐t② ❣✐✈❡s ❊ X n < ✵ ❊ X n > ✵ ❢♦r n < τ ❛♥❞ ❢♦r n ≥ τ. ▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✼ ✴

❚❍❊ ❈❯❙❯▼ ❚❊❙❚ ❢♦r ■✳■✳❉✳ ❉❆❚❆ ▲❡t S ✵ = ✵ ❛♥❞ ❧❡t n � ❢♦r n ≥ ✶ . S n = X k k = ✶ ❚❤❡♥ ❊ S n ❤❛s ❛ ♠✐♥✐♠✉♠ ❛t τ − ✶ . ❚❛s❦✿ ❛♣♣r♦①✐♠❛t❡ ♦♥✲❧✐♥❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ S n . ❚❤❡ ❈❯❙❯▼ st❛t✐st✐❝s ♦r P❛❣❡✲❍✐♥❦❧❡② ❞❡t❡❝t♦r✿ ❞❡✜♥❡ g n = S n − ♠✐♥ ✵ ≤ k ≤ n S k . ●❡♥❡r❛t❡ ❛♥ ❛❧❛r♠ ✐❢ g n > 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✉♣ P θ ✵ ( g n > δ ) . n Pr❛❝t✐❝❛❧ r❡❧❡✈❛♥❝❡✿ ❢❛❧s❡ ❛❧❛r♠ r❛t❡ ✭❋❆❘✮ ❞❡✜♥❡❞ ❛s ❛✳s✳ N ✶ � ❧✐♠ s✉♣ I { g n >δ } . N N n = ✶ Pr♦❜❧❡♠✿ ✜♥❞ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ❢♦r t❤❡ ❋❆❘✳ ▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✾ ✴

❚❍❊ ❉❨◆❆▼■❈❙ ♦❢ ❈❯❙❯▼ ❚❤❡ ❞②♥❛♠✐❝s ♦❢ g n ✐s ❡❛s✐❧② ♦❜t❛✐♥❡❞ ❛s ❢♦❧❧♦✇s✿ ✇✐t❤ g ✵ = ✵ . g n = ( g n − ✶ + X n ) + ❚❤✐s ❡st❛❜❧✐s❤❡s t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ q✉❡✉✐♥❣✱ W n ✱ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥✱ g n ✳ ❖❜❥❡❝t✐✈❡✿ ❜♦✉♥❞✐♥❣ t❤❡ ❡♠♣✐r✐❝❛❧ t❛✐❧ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ g n . ❚✇♦ r❡❧❛t❡❞ t❡❝❤♥✐❝❛❧ ♣r♦❜❧❡♠s✿ ❊①♣♦♥❡♥t✐❛❧ ❜♦✉♥❞s ❢♦r t❤❡ t❛✐❧ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ g n . ▼✐①✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ g n . ▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶✵ ✴

▼❖❚■❱❆❚■❖◆ ❢♦r ▼■❳■◆● ▲❡t ( ν n ) ❜❡ ❛♥ R s ✲✈❛❧✉❡❞ ✐✳✐✳❞✳ s❡q✉❡♥❝❡ ♦❢ r✳✈✳✲s s✉❝❤ t❤❛t E | ν n | q < + ∞ s✉♣ ✶ ≤ q < ∞ . ❢♦r ❛❧❧ n ≥ ✵ ▲❡t t❤❡ s × s ♠❛tr✐① A ❜❡ st❛❜❧❡✱ ❛♥❞ ❞❡✜♥❡ t❤❡ ✜❧t❡r❡❞ ♣r♦❝❡ss X ✵ = ✵ . X n = AX n − ✶ + ν n ✇✐t❤ ❉❡❝♦♠♣♦s❡ ( X n ) ❛s τ − ✶ � X n = A τ X n − τ + A k ν n − k . k = ✵ ▲✳ ●❡r❡♥❝sér ❆♥ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❡✐♥❣ ❛♥❞ ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ✶✶ ✴