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❖♣t✐♠❛❧ ▲♦ss❧❡ss ❙♦✉r❝❡ ❈♦❞❡s ❢♦r ❚✐♠❡❧② ❯♣❞❛t❡s Pr❛t❤❛♠❡s❤ ▼❛②❡❦❛r ❏♦✐♥t ✇♦r❦ ✇✐t❤ P❛r✐♠❛❧ P❛r❛❣ ❛♥❞ ❍✐♠❛♥s❤✉ ❚②❛❣✐ ❉❡♣❛rt♠❡♥t ♦❢ ❊❈❊✱ ■♥❞✐❛♥ ■♥st✐t✉t❡ ♦❢ ❙❝✐❡♥❝❡

❙♦✉r❝❡ ✲ ❚❤❡ ❍✐♥❞✉ ❚✐♠❡❧② ❯♣❞❛t❡s ❛r❡ ❝r✐t✐❝❛❧✳ ▼♦t✐✈❛t✐♦♥ ❙❡♥s♦r ❈❡♥t❡r ✶

❚✐♠❡❧② ❯♣❞❛t❡s ❛r❡ ❝r✐t✐❝❛❧✳ ▼♦t✐✈❛t✐♦♥ ❙❡♥s♦r ❈❡♥t❡r ❙♦✉r❝❡ ✲ ❚❤❡ ❍✐♥❞✉ ✶

▼♦t✐✈❛t✐♦♥ ❙❡♥s♦r ❈❡♥t❡r ❙♦✉r❝❡ ✲ ❚❤❡ ❍✐♥❞✉ ❚✐♠❡❧② ❯♣❞❛t❡s ❛r❡ ❝r✐t✐❝❛❧✳ ✶

▼♦t✐✈❛t✐♦♥ ❙❡♥s♦r ❈❡♥t❡r ❙♦✉r❝❡ ✲ ❚❤❡ ❍✐♥❞✉ ❚✐♠❡❧② ❯♣❞❛t❡s ❛r❡ ❝r✐t✐❝❛❧✳ ❆❣❡ ♦❢ ■♥❢♦r♠❛t✐♦♥ ✶ ✲ ♠❡tr✐❝ t♦ ❝❛♣t✉r❡ t✐♠❡❧✐♥❡ss✳ ✶ ❑❛✉❧✱ ❙✳✱ ❨❛t❡s✱ ❘✳✱ ❛♥❞ ●r✉t❡s❡r✱ ▼✳ ✭✷✵✶✶✱ ❉❡❝❡♠❜❡r✮✳ ❖♥ ♣✐❣❣②❜❛❝❦✐♥❣ ✐♥ ✈❡❤✐❝✉❧❛r ♥❡t✇♦r❦s✳ ■♥ ●❧♦❜❛❧ ❚❡❧❡❝♦♠♠✉♥✐❝❛t✐♦♥s ❈♦♥❢❡r❡♥❝❡ ✭●▲❖❇❊❈❖▼ ✶ ✷✵✶✶✮✱ ✷✵✶✶ ■❊❊❊ ✭♣♣✳ ✶✲✺✮✳ ■❊❊❊✳

❙❡♥s♦r ❈❡♥t❡r ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♠✐♥✐♠✐③✐♥❣ t❤❡ ❛✈❡r❛❣❡ ❛❣❡ ❲❡ r❡str✐❝t t♦ ▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s✳ ❆❣❡ ♦❢ ■♥❢♦r♠❛t✐♦♥ ✭❆❖■✮ ✲ ▼❡tr✐❝ ❢♦r ❚✐♠❡❧✐♥❡ss ◮ ❆❖■✿ ❚✐♠❡ ❧❛❣ ❜❡t✇❡❡♥ t❤❡ ❧❛t❡st ✐♥❢♦r♠❛t✐♦♥ ❛t t❤❡ ❘❳ ✇✳r✳t✳ t❤❛t ❛t ❚❳✳ ✷

❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♠✐♥✐♠✐③✐♥❣ t❤❡ ❛✈❡r❛❣❡ ❛❣❡ ❲❡ r❡str✐❝t t♦ ▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s✳ ❆❣❡ ♦❢ ■♥❢♦r♠❛t✐♦♥ ✭❆❖■✮ ✲ ▼❡tr✐❝ ❢♦r ❚✐♠❡❧✐♥❡ss ◮ ❆❖■✿ ❚✐♠❡ ❧❛❣ ❜❡t✇❡❡♥ t❤❡ ❧❛t❡st ✐♥❢♦r♠❛t✐♦♥ ❛t t❤❡ ❘❳ ✇✳r✳t✳ t❤❛t ❛t ❚❳✳ ❙❡♥s♦r ❈❡♥t❡r ✷

❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♠✐♥✐♠✐③✐♥❣ t❤❡ ❛✈❡r❛❣❡ ❛❣❡ ❲❡ r❡str✐❝t t♦ ▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s✳ ❆❣❡ ♦❢ ■♥❢♦r♠❛t✐♦♥ ✭❆❖■✮ ✲ ▼❡tr✐❝ ❢♦r ❚✐♠❡❧✐♥❡ss ◮ ❆❖■✿ ❚✐♠❡ ❧❛❣ ❜❡t✇❡❡♥ t❤❡ ❧❛t❡st ✐♥❢♦r♠❛t✐♦♥ ❛t t❤❡ ❘❳ ✇✳r✳t✳ t❤❛t ❛t ❚❳✳ X t ❙❡♥s♦r ❈❡♥t❡r ✷

❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♠✐♥✐♠✐③✐♥❣ t❤❡ ❛✈❡r❛❣❡ ❛❣❡ ❲❡ r❡str✐❝t t♦ ▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s✳ ❆❣❡ ♦❢ ■♥❢♦r♠❛t✐♦♥ ✭❆❖■✮ ✲ ▼❡tr✐❝ ❢♦r ❚✐♠❡❧✐♥❡ss ◮ ❆❖■✿ ❚✐♠❡ ❧❛❣ ❜❡t✇❡❡♥ t❤❡ ❧❛t❡st ✐♥❢♦r♠❛t✐♦♥ ❛t t❤❡ ❘❳ ✇✳r✳t✳ t❤❛t ❛t ❚❳✳ X U ( t ) X t ❙❡♥s♦r ❈❡♥t❡r ✷

❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♠✐♥✐♠✐③✐♥❣ t❤❡ ❛✈❡r❛❣❡ ❛❣❡ ❲❡ r❡str✐❝t t♦ ▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s✳ ❆❣❡ ♦❢ ■♥❢♦r♠❛t✐♦♥ ✭❆❖■✮ ✲ ▼❡tr✐❝ ❢♦r ❚✐♠❡❧✐♥❡ss ◮ ❆❖■✿ ❚✐♠❡ ❧❛❣ ❜❡t✇❡❡♥ t❤❡ ❧❛t❡st ✐♥❢♦r♠❛t✐♦♥ ❛t t❤❡ ❘❳ ✇✳r✳t✳ t❤❛t ❛t ❚❳✳ X U ( t ) X t ❙❡♥s♦r ❈❡♥t❡r A ( t ) = t − U ( t ) . ✷

❲❡ r❡str✐❝t t♦ ▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s✳ ❆❣❡ ♦❢ ■♥❢♦r♠❛t✐♦♥ ✭❆❖■✮ ✲ ▼❡tr✐❝ ❢♦r ❚✐♠❡❧✐♥❡ss ◮ ❆❖■✿ ❚✐♠❡ ❧❛❣ ❜❡t✇❡❡♥ t❤❡ ❧❛t❡st ✐♥❢♦r♠❛t✐♦♥ ❛t t❤❡ ❘❳ ✇✳r✳t✳ t❤❛t ❛t ❚❳✳ X U ( t ) X t ❙❡♥s♦r ❈❡♥t❡r A ( t ) = t − U ( t ) . ◮ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♠✐♥✐♠✐③✐♥❣ t❤❡ ❛✈❡r❛❣❡ ❛❣❡ T 1 ¯ � A � lim sup A ( t ) . T T →∞ t =1 ✷

❆❣❡ ♦❢ ■♥❢♦r♠❛t✐♦♥ ✭❆❖■✮ ✲ ▼❡tr✐❝ ❢♦r ❚✐♠❡❧✐♥❡ss ◮ ❆❖■✿ ❚✐♠❡ ❧❛❣ ❜❡t✇❡❡♥ t❤❡ ❧❛t❡st ✐♥❢♦r♠❛t✐♦♥ ❛t t❤❡ ❘❳ ✇✳r✳t✳ t❤❛t ❛t ❚❳✳ X U ( t ) X t ❙❡♥s♦r ❈❡♥t❡r A ( t ) = t − U ( t ) . ◮ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♠✐♥✐♠✐③✐♥❣ t❤❡ ❛✈❡r❛❣❡ ❛❣❡ T 1 ¯ � A � lim sup A ( t ) . T T →∞ t =1 ◮ ❲❡ r❡str✐❝t t♦ ▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s✳ ✷

❚✐♠❡ ▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s ❙♦✉r❝❡ ❊♥❝♦❞❡r ❈❤❛♥♥❡❧ ❉❡❝♦❞❡r Pr❡✜①✲❢r❡❡ ◆♦✐s❡❧❡ss iid P ✶ ❇✐t✴❚✐♠❡ s❧♦t ✸

▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s ❚✐♠❡ X 1 e ( X 1 ) ❙♦✉r❝❡ ❊♥❝♦❞❡r ❈❤❛♥♥❡❧ ❉❡❝♦❞❡r Pr❡✜①✲❢r❡❡ ◆♦✐s❡❧❡ss iid P ✶ ❇✐t✴❚✐♠❡ s❧♦t ✸

▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s ❚✐♠❡ X 2 X 1 e ( X 1 ) ❙♦✉r❝❡ ❊♥❝♦❞❡r ❈❤❛♥♥❡❧ ❉❡❝♦❞❡r Pr❡✜①✲❢r❡❡ ◆♦✐s❡❧❡ss iid P ✶ ❇✐t✴❚✐♠❡ s❧♦t ✸

▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s ❚✐♠❡ e ( X 3 ) X 3 X 1 X 2 X 1 e ( X 1 ) ❙♦✉r❝❡ ❊♥❝♦❞❡r ❈❤❛♥♥❡❧ ❉❡❝♦❞❡r Pr❡✜①✲❢r❡❡ ◆♦✐s❡❧❡ss iid P ✶ ❇✐t✴❚✐♠❡ s❧♦t ✸

▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s e ( X 4 ) X 4 X 3 ❚✐♠❡ e ( X 3 ) X 3 X 1 X 2 X 1 e ( X 1 ) ❙♦✉r❝❡ ❊♥❝♦❞❡r ❈❤❛♥♥❡❧ ❉❡❝♦❞❡r Pr❡✜①✲❢r❡❡ ◆♦✐s❡❧❡ss iid P ✶ ❇✐t✴❚✐♠❡ s❧♦t ✸

▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s X 5 X 3 e ( X 4 ) X 4 X 3 ❚✐♠❡ e ( X 3 ) X 3 X 1 X 2 X 1 e ( X 1 ) ❙♦✉r❝❡ ❊♥❝♦❞❡r ❈❤❛♥♥❡❧ ❉❡❝♦❞❡r Pr❡✜①✲❢r❡❡ ◆♦✐s❡❧❡ss iid P ✶ ❇✐t✴❚✐♠❡ s❧♦t ✸

▼❡♠♦r②❧❡ss ❯♣❞❛t❡ ❙❝❤❡♠❡s X 6 X 3 X 5 X 3 e ( X 4 ) X 4 X 3 ❚✐♠❡ e ( X 3 ) X 3 X 1 X 2 X 1 e ( X 1 ) ❙♦✉r❝❡ ❊♥❝♦❞❡r ❈❤❛♥♥❡❧ ❉❡❝♦❞❡r Pr❡✜①✲❢r❡❡ ◆♦✐s❡❧❡ss iid P ✶ ❇✐t✴❚✐♠❡ s❧♦t ✸

■❧❧✉str❛t✐♦♥ ♦❢ ■♥st❛♥t❛♥❡♦✉s ❆❣❡ X 6 X 3 X 5 X 3 X 4 e ( X 4 ) X 3 ❚✐♠❡ e ( X 3 ) X 3 X 1 X 2 X 1 e ( X 1 ) ❙♦✉r❝❡ ❊♥❝♦❞❡r ❈❤❛♥♥❡❧ ❉❡❝♦❞❡r ✹✰ ✸✰ A ( t ) A ( t ) = t − U ( t ) ✷ ✰ U ( t ) = ■♥❞❡① ♦❢ ❧❛t❡st ✐♥❢♦r♠❛t✐♦♥ ❛t t❤❡ ❞❡❝♦❞❡r ✶✰ ✵ ✰ ✰ ✰ ✰ ✰ ✶ ✷ ✸ ✹ ✺ ✻ ✹ t

❚❤❡♦r❡♠ ❋♦r ❛ ♣r❡✜①✲❢r❡❡ ❝♦❞❡ ✱ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❆✈❡r❛❣❡ ❆❣❡ T 1 ¯ � A ( e ) � lim sup A ( t ) T T →∞ t =1 ℓ ( x ) � ❝♦❞❡✲❧❡♥❣t❤ ❢♦r ❛ s②♠❜♦❧ x ✱ L � ℓ ( X ) ✳ ✺

❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❆✈❡r❛❣❡ ❆❣❡ T 1 ¯ � A ( e ) � lim sup A ( t ) T T →∞ t =1 ℓ ( x ) � ❝♦❞❡✲❧❡♥❣t❤ ❢♦r ❛ s②♠❜♦❧ x ✱ L � ℓ ( X ) ✳ ❚❤❡♦r❡♠ E [ L 2 ] ❋♦r ❛ ♣r❡✜①✲❢r❡❡ ❝♦❞❡ e ✱ ¯ 2 E [ L ] − 1 A ( e ) = E [ L ] + a.s.. 2 ✺

❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❆✈❡r❛❣❡ ❆❣❡ T 1 ¯ � A ( e ) � lim sup A ( t ) T T →∞ t =1 ℓ ( x ) � ❝♦❞❡✲❧❡♥❣t❤ ❢♦r ❛ s②♠❜♦❧ x ✱ L � ℓ ( X ) ✳ ❚❤❡♦r❡♠ E [ L 2 ] ❋♦r ❛ ♣r❡✜①✲❢r❡❡ ❝♦❞❡ e ✱ ¯ 2 E [ L ] − 1 A ( e ) = E [ L ] + a.s.. 2 Pr♦♦❢ ■❞❡❛✿ ◮ S i � i th r❡❝❡♣t✐♦♥ ✹✰ ✸✰ A ( t ) ✷ ✰ ✶✰ ✵ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ S 0 S 1 S 2 S 3 S 4 S 5 ✺