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tr tt sqs r rr t t trs s

  1. ❱❡❝t♦r✲✈❛❧✉❡❞ t❛♥❣❡♥t s❡q✉❡♥❝❡s ❛♥❞ ❞❡❝♦✉♣❧✐♥❣✳ ▼❛r❦ ❱❡r❛❛r ❉❡❧❢t ✭t❤❡ ◆❡t❤❡r❧❛♥❞s✮ ❆❞✈❛♥❝❡s ✐♥ ❙t♦❝❤❛st✐❝ ■♥❡q✉❛❧✐t✐❡s ❛♥❞ t❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s ❏✉♥❡ ✼✲✶✷✱ ✷✵✵✾ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❙♦♥❥❛ ❈♦① ▼❛r❦ ❱❡r❛❛r ✭❉❡❧❢t ✭t❤❡ ◆❡t❤❡r❧❛♥❞s✮✮ ❚❛♥❣❡♥t s❡q✉❡♥❝❡s ❛♥❞ ❞❡❝♦✉♣❧✐♥❣ ✶ ✴ ✶✸

  2. ❚❤❡ ❛♥s✇❡r t♦ ❛❧❧ ✐s ②❡s ❛♥❞ ✐s ❦♥♦✇♥ ❢♦r ♠♦r❡ t❤❛♥ ✷✵ ②❡❛rs✳ ❚❤❡ ✉s❡❢✉❧♥❡ss ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ♥ ✈ ♥ ♥ ✶ ✐s ❡❛s✐❡r t♦ ❛♥❛❧②③❡✳ ❖♥❧② ♣ ✶ ✳ ❚❤❡r❡ ✐s ♥♦ t❛✐❧ ❞♦♠✐♥❛t✐♦♥ ❳ t ❝ ✶ ❨ ❝ ✷ t ✳ ■♥tr♦❞✉❝t✐♦♥ ▲❡t (Ω , A , ( F ♥ ) ♥ ≥ ✵ , P ) ❜❡ ✜❧t❡r❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❊①❛♠♣❧❡ ✭r❡❛❧ ❝❛s❡✮ ( ξ ♥ ) ♥ ≥ ✶ ✲ ❛❞❛♣t❡❞ ✐♥❞❡♣❡♥❞❡♥t s②♠♠❡tr✐❝ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ( ✈ ♥ ) ♥ ≥ ✶ ✲ ♣r❡❞✐❝t❛❜❧❡ s❡q✉❡♥❝❡✳ ( ξ ′ ♥ ) ♥ ≥ ✶ ✲ ✐♥❞❡♣❡♥❞❡♥t ❝♦♣② ♦❢ ( ξ ♥ ) ♥ ≥ ✶ ✳ ❈❛♥ ✇❡ ❝♦♠♣❛r❡ ❳ = � ◆ ♥ = ✶ ξ ♥ ✈ ♥ ❛♥❞ ❨ = � ◆ ♥ = ✶ ξ ′ ♥ ✈ ♥ ✐♥ ▲ ♣ ❄ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐❧✐t② ❢♦r ◆ → ∞ ❄ ❛❧♠♦st s✉r❡ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r ◆ → ∞ ❄ ▼❛r❦ ❱❡r❛❛r ✭❉❡❧❢t ✭t❤❡ ◆❡t❤❡r❧❛♥❞s✮✮ ❚❛♥❣❡♥t s❡q✉❡♥❝❡s ❛♥❞ ❞❡❝♦✉♣❧✐♥❣ ✷ ✴ ✶✸

  3. ❖♥❧② ♣ ✶ ✳ ❚❤❡r❡ ✐s ♥♦ t❛✐❧ ❞♦♠✐♥❛t✐♦♥ ❳ t ❝ ✶ ❨ ❝ ✷ t ✳ ■♥tr♦❞✉❝t✐♦♥ ▲❡t (Ω , A , ( F ♥ ) ♥ ≥ ✵ , P ) ❜❡ ✜❧t❡r❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❊①❛♠♣❧❡ ✭r❡❛❧ ❝❛s❡✮ ( ξ ♥ ) ♥ ≥ ✶ ✲ ❛❞❛♣t❡❞ ✐♥❞❡♣❡♥❞❡♥t s②♠♠❡tr✐❝ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ( ✈ ♥ ) ♥ ≥ ✶ ✲ ♣r❡❞✐❝t❛❜❧❡ s❡q✉❡♥❝❡✳ ( ξ ′ ♥ ) ♥ ≥ ✶ ✲ ✐♥❞❡♣❡♥❞❡♥t ❝♦♣② ♦❢ ( ξ ♥ ) ♥ ≥ ✶ ✳ ❈❛♥ ✇❡ ❝♦♠♣❛r❡ ❳ = � ◆ ♥ = ✶ ξ ♥ ✈ ♥ ❛♥❞ ❨ = � ◆ ♥ = ✶ ξ ′ ♥ ✈ ♥ ✐♥ ▲ ♣ ❄ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐❧✐t② ❢♦r ◆ → ∞ ❄ ❛❧♠♦st s✉r❡ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r ◆ → ∞ ❄ ❚❤❡ ❛♥s✇❡r t♦ ❛❧❧ ✐s ②❡s ❛♥❞ ✐s ❦♥♦✇♥ ❢♦r ♠♦r❡ t❤❛♥ ✷✵ ②❡❛rs✳ ❚❤❡ ✉s❡❢✉❧♥❡ss ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ( ξ ′ ♥ ✈ ♥ ) ♥ ≥ ✶ ✐s ❡❛s✐❡r t♦ ❛♥❛❧②③❡✳ ▼❛r❦ ❱❡r❛❛r ✭❉❡❧❢t ✭t❤❡ ◆❡t❤❡r❧❛♥❞s✮✮ ❚❛♥❣❡♥t s❡q✉❡♥❝❡s ❛♥❞ ❞❡❝♦✉♣❧✐♥❣ ✷ ✴ ✶✸

  4. ■♥tr♦❞✉❝t✐♦♥ ▲❡t (Ω , A , ( F ♥ ) ♥ ≥ ✵ , P ) ❜❡ ✜❧t❡r❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❊①❛♠♣❧❡ ✭r❡❛❧ ❝❛s❡✮ ( ξ ♥ ) ♥ ≥ ✶ ✲ ❛❞❛♣t❡❞ ✐♥❞❡♣❡♥❞❡♥t s②♠♠❡tr✐❝ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ( ✈ ♥ ) ♥ ≥ ✶ ✲ ♣r❡❞✐❝t❛❜❧❡ s❡q✉❡♥❝❡✳ ( ξ ′ ♥ ) ♥ ≥ ✶ ✲ ✐♥❞❡♣❡♥❞❡♥t ❝♦♣② ♦❢ ( ξ ♥ ) ♥ ≥ ✶ ✳ ❈❛♥ ✇❡ ❝♦♠♣❛r❡ ❳ = � ◆ ♥ = ✶ ξ ♥ ✈ ♥ ❛♥❞ ❨ = � ◆ ♥ = ✶ ξ ′ ♥ ✈ ♥ ✐♥ ▲ ♣ ❄ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ♣r♦❜❛❜✐❧✐t② ❢♦r ◆ → ∞ ❄ ❛❧♠♦st s✉r❡ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r ◆ → ∞ ❄ ❚❤❡ ❛♥s✇❡r t♦ ❛❧❧ ✐s ②❡s ❛♥❞ ✐s ❦♥♦✇♥ ❢♦r ♠♦r❡ t❤❛♥ ✷✵ ②❡❛rs✳ ❚❤❡ ✉s❡❢✉❧♥❡ss ❝♦♠❡s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ( ξ ′ ♥ ✈ ♥ ) ♥ ≥ ✶ ✐s ❡❛s✐❡r t♦ ❛♥❛❧②③❡✳ ❖♥❧② ♣ ∈ ( ✶ , ∞ ) ✳ ❚❤❡r❡ ✐s ♥♦ t❛✐❧ ❞♦♠✐♥❛t✐♦♥ P ( | ❳ | > t ) ≤ ❝ ✶ P ( | ❨ | > ❝ ✷ t ) ✳ ▼❛r❦ ❱❡r❛❛r ✭❉❡❧❢t ✭t❤❡ ◆❡t❤❡r❧❛♥❞s✮✮ ❚❛♥❣❡♥t s❡q✉❡♥❝❡s ❛♥❞ ❞❡❝♦✉♣❧✐♥❣ ✷ ✴ ✶✸

  5. ❙♦♠❡ ♦❢ t❤❡ ❛✉t❤♦rs ✇❤✐❝❤ ✇♦r❦❡❞ ♦♥ t❛♥❣❡♥t s❡q✉❡♥❝❡s ❛♥❞ r❡❧❛t❡❞ ✜❡❧❞s✿ ❇✉r❦❤♦❧❞❡r✱ ❞❡ ❧❛ P❡ñ❛✱ ●❛r❧✐♥❣✱ ●❡✐ss✱ ●✐♥é✱ ❍✐t❝③❡♥❦♦✱ ❏❛❝♦❞✱ ❏❛❦✉❜♦✇❦s✐✱ ❑❧❛ss✱ ❑✇❛♣✐❡➠✱ ▼❝❈♦♥♥❡❧❧✱ ▼♦♥t❣♦♠❡r②✲❙♠✐t❤✱ P✐❛s❡❝❦✐✱ ❘♦s✐➠s❦✐✱ ❲♦②❝③②➠s❦✐✱ ❩✐♥♥✱ ✳✳✳✳✳ ❚❤❡ ❜♦♦❦s ❑✇❛♣✐❡➠✕❲♦②❝③②➠s❦✐ ✭✶✾✾✷✮ ❛♥❞ ❞❡ ❧❛ P❡ñ❛✕●✐♥é ✭✶✾✾✾✮ ❛r❡ ❣♦♦❞ r❡❢❡r❡♥❝❡s ❢♦r t❛♥❣❡♥t s❡q✉❡♥❝❡s✳ ❩✐♥♥ ✭✶✾✽✺✮✿ ❢♦r ♣ ✷✱ ❜② t❤❡ ❇✉r❦❤♦❧❞❡r✲❘♦s❡♥t❤❛❧ ✐♥❡q✉❛❧✐t✐❡s ◆ ◆ ◆ ✶ ✶ ✷ ✷ ♣ ♣ ❞ ♥ ❞ ♥ ❞ ♥ ♣ ♥ ✶ ♣ ♣ ♣ ♥ ✶ ♥ ✶ ♥ ✶ ◆ ◆ ◆ ✶ ✶ ✷ ✷ ♣ ♣ ❡ ♥ ❡ ♥ ❡ ♥ ♥ ✶ ♣ ♣ ♣ ♣ ♥ ✶ ♥ ✶ ♥ ✶ ▲❡t ❞ ♥ = ξ ♥ ✈ ♥ ❛♥❞ ❡ ♥ = ξ ′ ♥ ✈ ♥ ✳ ❚❤❡ s❡q✉❡♥❝❡s ( ❞ ♥ ) ♥ ≥ ✶ ❛♥❞ ( ❡ ♥ ) ♥ ≥ ✶ ✜t ✐♥ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ ❛♥❞ ❛r❡ ❛♥ ✐♠♣♦rt❛♥t ❡①❛♠♣❧❡ ♦❢ s♦✲❝❛❧❧❡❞ t❛♥❣❡♥t ♠❛rt✐♥❣❛❧❡ ❞✐✛❡r❡♥❝❡ s❡q✉❡♥❝❡s✳ ▼♦r❡♦✈❡r✱ ( ❡ ♥ ) ♥ ≥ ✶ ✐s ❛ s♦✲❝❛❧❧❡❞ ❞❡❝♦✉♣❧❡❞ t❛♥❣❡♥t s❡q✉❡♥❝❡ ♦❢ ( ❞ ♥ ) ♥ ≥ ✶ ✳ ▼❛r❦ ❱❡r❛❛r ✭❉❡❧❢t ✭t❤❡ ◆❡t❤❡r❧❛♥❞s✮✮ ❚❛♥❣❡♥t s❡q✉❡♥❝❡s ❛♥❞ ❞❡❝♦✉♣❧✐♥❣ ✸ ✴ ✶✸

  6. ❩✐♥♥ ✭✶✾✽✺✮✿ ❢♦r ♣ ✷✱ ❜② t❤❡ ❇✉r❦❤♦❧❞❡r✲❘♦s❡♥t❤❛❧ ✐♥❡q✉❛❧✐t✐❡s ◆ ◆ ◆ ✶ ✶ ✷ ✷ ♣ ♣ ❞ ♥ ❞ ♥ ❞ ♥ ♣ ♥ ✶ ♣ ♣ ♣ ♥ ✶ ♥ ✶ ♥ ✶ ◆ ◆ ◆ ✶ ✶ ✷ ✷ ♣ ♣ ❡ ♥ ❡ ♥ ❡ ♥ ♥ ✶ ♣ ♣ ♣ ♣ ♥ ✶ ♥ ✶ ♥ ✶ ▲❡t ❞ ♥ = ξ ♥ ✈ ♥ ❛♥❞ ❡ ♥ = ξ ′ ♥ ✈ ♥ ✳ ❚❤❡ s❡q✉❡♥❝❡s ( ❞ ♥ ) ♥ ≥ ✶ ❛♥❞ ( ❡ ♥ ) ♥ ≥ ✶ ✜t ✐♥ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ ❛♥❞ ❛r❡ ❛♥ ✐♠♣♦rt❛♥t ❡①❛♠♣❧❡ ♦❢ s♦✲❝❛❧❧❡❞ t❛♥❣❡♥t ♠❛rt✐♥❣❛❧❡ ❞✐✛❡r❡♥❝❡ s❡q✉❡♥❝❡s✳ ▼♦r❡♦✈❡r✱ ( ❡ ♥ ) ♥ ≥ ✶ ✐s ❛ s♦✲❝❛❧❧❡❞ ❞❡❝♦✉♣❧❡❞ t❛♥❣❡♥t s❡q✉❡♥❝❡ ♦❢ ( ❞ ♥ ) ♥ ≥ ✶ ✳ ❙♦♠❡ ♦❢ t❤❡ ❛✉t❤♦rs ✇❤✐❝❤ ✇♦r❦❡❞ ♦♥ t❛♥❣❡♥t s❡q✉❡♥❝❡s ❛♥❞ r❡❧❛t❡❞ ✜❡❧❞s✿ ❇✉r❦❤♦❧❞❡r✱ ❞❡ ❧❛ P❡ñ❛✱ ●❛r❧✐♥❣✱ ●❡✐ss✱ ●✐♥é✱ ❍✐t❝③❡♥❦♦✱ ❏❛❝♦❞✱ ❏❛❦✉❜♦✇❦s✐✱ ❑❧❛ss✱ ❑✇❛♣✐❡➠✱ ▼❝❈♦♥♥❡❧❧✱ ▼♦♥t❣♦♠❡r②✲❙♠✐t❤✱ P✐❛s❡❝❦✐✱ ❘♦s✐➠s❦✐✱ ❲♦②❝③②➠s❦✐✱ ❩✐♥♥✱ ✳✳✳✳✳ ❚❤❡ ❜♦♦❦s ❑✇❛♣✐❡➠✕❲♦②❝③②➠s❦✐ ✭✶✾✾✷✮ ❛♥❞ ❞❡ ❧❛ P❡ñ❛✕●✐♥é ✭✶✾✾✾✮ ❛r❡ ❣♦♦❞ r❡❢❡r❡♥❝❡s ❢♦r t❛♥❣❡♥t s❡q✉❡♥❝❡s✳ ▼❛r❦ ❱❡r❛❛r ✭❉❡❧❢t ✭t❤❡ ◆❡t❤❡r❧❛♥❞s✮✮ ❚❛♥❣❡♥t s❡q✉❡♥❝❡s ❛♥❞ ❞❡❝♦✉♣❧✐♥❣ ✸ ✴ ✶✸

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