❘❛r❡ ❡✈❡♥t s✐♠✉❧❛t✐♦♥ ❛ P♦✐♥t Pr♦❝❡ss ✐♥t❡r♣r❡t❛t✐♦♥ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ♣r♦❜❛❜✐❧✐t② ❛♥❞ q✉❛♥t✐❧❡ ❡st✐♠❛t✐♦♥ ❛♥❞ ♠❡t❛♠♦❞❡❧ ❜❛s❡❞ ❛❧❣♦r✐t❤♠s ❙é♠✐♥❛✐r❡ ❙ 3 | ❈❧é♠❡♥t ❲❆▲❚❊❘ ▼❛r❝❤ ✶✸t❤ ✷✵✶✺
■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ s❡tt✐♥❣✿ X r❛♥❞♦♠ ✈❡❝t♦r ✇✐t❤ ❦♥♦✇ ❞✐str✐❜✉t✐♦♥ µ X g ❛ ✧❜❧❛❝❦✲❜♦①✧ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥t✐♥❣ ❛ ❝♦♠♣✉t❡r ❝♦❞❡✿ g : R d → R Y = g ( X ) t❤❡ r❡❛❧✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇❤✐❝❤ ❞❡s❝r✐❜❡s t❤❡ st❛t❡ ♦❢ t❤❡ s②st❡♠❀ ✐ts ❞✐str✐❜✉t✐♦♥ µ Y ✐s ✉♥❦♥♦✇♥ ❯♥❝❡rt❛✐♥t② ◗✉❛♥t✐✜❝❛t✐♦♥✿ F = { x ∈ R d | g( x ) > q } ✜♥❞ p = P [ X ∈ F ] = µ X ( F ) ❢♦r ❛ ❣✐✈❡♥ q ✜♥❞ q ❢♦r ❛ ❣✐✈❡♥ p ■ss✉❡s p = µ X ( F ) = µ Y ([ q ; + ∞ [) ≪ 1 ♥❡❡❞s t♦ ✉s❡ g t♦ ❣❡t F ♦r µ Y ✇❤✐❝❤ ✐s t✐♠❡ ❝♦st❧② ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t♦r ❤❛s ❛ ❈❱ δ 2 ≈ 1 /Np ⇒ N ≫ 1 /p ❙é♠✐♥❛✐r❡ ❙ 3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✴✷✻
■♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ▼✉❧t✐❧❡✈❡❧ s♣❧✐tt✐♥❣ ▼✉❧t✐❧❡✈❡❧ ❙♣❧✐tt✐♥❣ ✭ ❙✉❜s❡t ❙✐♠✉❧❛t✐♦♥s✮ ❲r✐t❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t② ❛s ❛ ♣r♦❞✉❝t ♦❢ ❧❡ss s♠❛❧❧ ♣r♦❜❛❜✐❧✐t✐❡s ❛♥❞ ❡st✐♠❛t❡ t❤❡♠ ✇✐t❤ ▼❈▼❈✿ ❧❡t ❜❡ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ✇✐t❤ ❛♥❞ ✿ ❤♦✇ t♦ ❝❤♦♦s❡ ✿ ❜❡❢♦r❡❤❛♥❞ ♦r ♦♥✲t❤❡✲❣♦❄ ❖♣t✐♠❛❧✐t②❄ ❇✐❛s❄ ■♥tr♦❞✉❝t✐♦♥ ❚✇♦ ♠❛✐♥ ❞✐r❡❝t✐♦♥s t♦ ♦✈❡r❝♦♠❡ t❤✐s ✐ss✉❡✿ ❧❡❛r♥ ❛ ♠❡t❛♠♦❞❡❧ ♦♥ g ✉s❡ ✈❛r✐❛♥❝❡✲r❡❞✉❝t✐♦♥ t❡❝❤♥✐q✉❡s t♦ ❡st✐♠❛t❡ p ❙é♠✐♥❛✐r❡ ❙ 3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✴✷✻
■♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ▼✉❧t✐❧❡✈❡❧ s♣❧✐tt✐♥❣ ▼✉❧t✐❧❡✈❡❧ ❙♣❧✐tt✐♥❣ ✭ ❙✉❜s❡t ❙✐♠✉❧❛t✐♦♥s✮ ❲r✐t❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t② ❛s ❛ ♣r♦❞✉❝t ♦❢ ❧❡ss s♠❛❧❧ ♣r♦❜❛❜✐❧✐t✐❡s ❛♥❞ ❡st✐♠❛t❡ t❤❡♠ ✇✐t❤ ▼❈▼❈✿ ❧❡t ❜❡ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ✇✐t❤ ❛♥❞ ✿ ❤♦✇ t♦ ❝❤♦♦s❡ ✿ ❜❡❢♦r❡❤❛♥❞ ♦r ♦♥✲t❤❡✲❣♦❄ ❖♣t✐♠❛❧✐t②❄ ❇✐❛s❄ ■♥tr♦❞✉❝t✐♦♥ ❚✇♦ ♠❛✐♥ ❞✐r❡❝t✐♦♥s t♦ ♦✈❡r❝♦♠❡ t❤✐s ✐ss✉❡✿ ❧❡❛r♥ ❛ ♠❡t❛♠♦❞❡❧ ♦♥ g ✉s❡ ✈❛r✐❛♥❝❡✲r❡❞✉❝t✐♦♥ t❡❝❤♥✐q✉❡s t♦ ❡st✐♠❛t❡ p ❙é♠✐♥❛✐r❡ ❙ 3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✴✷✻
▼✉❧t✐❧❡✈❡❧ s♣❧✐tt✐♥❣ ▼✉❧t✐❧❡✈❡❧ ❙♣❧✐tt✐♥❣ ✭ ❙✉❜s❡t ❙✐♠✉❧❛t✐♦♥s✮ ❲r✐t❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t② ❛s ❛ ♣r♦❞✉❝t ♦❢ ❧❡ss s♠❛❧❧ ♣r♦❜❛❜✐❧✐t✐❡s ❛♥❞ ❡st✐♠❛t❡ t❤❡♠ ✇✐t❤ ▼❈▼❈✿ ❧❡t ❜❡ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ✇✐t❤ ❛♥❞ ✿ ❤♦✇ t♦ ❝❤♦♦s❡ ✿ ❜❡❢♦r❡❤❛♥❞ ♦r ♦♥✲t❤❡✲❣♦❄ ❖♣t✐♠❛❧✐t②❄ ❇✐❛s❄ ■♥tr♦❞✉❝t✐♦♥ ❚✇♦ ♠❛✐♥ ❞✐r❡❝t✐♦♥s t♦ ♦✈❡r❝♦♠❡ t❤✐s ✐ss✉❡✿ ❧❡❛r♥ ❛ ♠❡t❛♠♦❞❡❧ ♦♥ g ✉s❡ ✈❛r✐❛♥❝❡✲r❡❞✉❝t✐♦♥ t❡❝❤♥✐q✉❡s t♦ ❡st✐♠❛t❡ p ■♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ❙é♠✐♥❛✐r❡ ❙ 3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✴✷✻
▼✉❧t✐❧❡✈❡❧ ❙♣❧✐tt✐♥❣ ✭ ❙✉❜s❡t ❙✐♠✉❧❛t✐♦♥s✮ ❲r✐t❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t② ❛s ❛ ♣r♦❞✉❝t ♦❢ ❧❡ss s♠❛❧❧ ♣r♦❜❛❜✐❧✐t✐❡s ❛♥❞ ❡st✐♠❛t❡ t❤❡♠ ✇✐t❤ ▼❈▼❈✿ ❧❡t ❜❡ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ✇✐t❤ ❛♥❞ ✿ ❤♦✇ t♦ ❝❤♦♦s❡ ✿ ❜❡❢♦r❡❤❛♥❞ ♦r ♦♥✲t❤❡✲❣♦❄ ❖♣t✐♠❛❧✐t②❄ ❇✐❛s❄ ■♥tr♦❞✉❝t✐♦♥ ❚✇♦ ♠❛✐♥ ❞✐r❡❝t✐♦♥s t♦ ♦✈❡r❝♦♠❡ t❤✐s ✐ss✉❡✿ ❧❡❛r♥ ❛ ♠❡t❛♠♦❞❡❧ ♦♥ g ✉s❡ ✈❛r✐❛♥❝❡✲r❡❞✉❝t✐♦♥ t❡❝❤♥✐q✉❡s t♦ ❡st✐♠❛t❡ p ■♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ▼✉❧t✐❧❡✈❡❧ s♣❧✐tt✐♥❣ ❙é♠✐♥❛✐r❡ ❙ 3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✴✷✻
❤♦✇ t♦ ❝❤♦♦s❡ ✿ ❜❡❢♦r❡❤❛♥❞ ♦r ♦♥✲t❤❡✲❣♦❄ ❖♣t✐♠❛❧✐t②❄ ❇✐❛s❄ ■♥tr♦❞✉❝t✐♦♥ ❚✇♦ ♠❛✐♥ ❞✐r❡❝t✐♦♥s t♦ ♦✈❡r❝♦♠❡ t❤✐s ✐ss✉❡✿ ❧❡❛r♥ ❛ ♠❡t❛♠♦❞❡❧ ♦♥ g ✉s❡ ✈❛r✐❛♥❝❡✲r❡❞✉❝t✐♦♥ t❡❝❤♥✐q✉❡s t♦ ❡st✐♠❛t❡ p ■♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ▼✉❧t✐❧❡✈❡❧ s♣❧✐tt✐♥❣ ▼✉❧t✐❧❡✈❡❧ ❙♣❧✐tt✐♥❣ ✭ ❙✉❜s❡t ❙✐♠✉❧❛t✐♦♥s✮ ❲r✐t❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t② p ❛s ❛ ♣r♦❞✉❝t ♦❢ ❧❡ss s♠❛❧❧ ♣r♦❜❛❜✐❧✐t✐❡s ❛♥❞ ❡st✐♠❛t❡ t❤❡♠ ✇✐t❤ ▼❈▼❈✿ ❧❡t ( q i ) i =0 ..m ❜❡ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ✇✐t❤ q 0 = −∞ ❛♥❞ q m = q ✿ p = P [ g ( X ) > q ] = P [ g ( X ) > q m | g ( X ) > q m − 1 ] × P [ g ( X ) > q m − 1 ] = P [ g ( X ) > q ] = P [ g ( X ) > q m | g ( X ) > q m − 1 ] × · · · × P [ g ( X ) > q 1 ] ❙é♠✐♥❛✐r❡ ❙ 3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✴✷✻
■♥tr♦❞✉❝t✐♦♥ ❚✇♦ ♠❛✐♥ ❞✐r❡❝t✐♦♥s t♦ ♦✈❡r❝♦♠❡ t❤✐s ✐ss✉❡✿ ❧❡❛r♥ ❛ ♠❡t❛♠♦❞❡❧ ♦♥ g ✉s❡ ✈❛r✐❛♥❝❡✲r❡❞✉❝t✐♦♥ t❡❝❤♥✐q✉❡s t♦ ❡st✐♠❛t❡ p ■♠♣♦rt❛♥❝❡ s❛♠♣❧✐♥❣ ▼✉❧t✐❧❡✈❡❧ s♣❧✐tt✐♥❣ ▼✉❧t✐❧❡✈❡❧ ❙♣❧✐tt✐♥❣ ✭ ❙✉❜s❡t ❙✐♠✉❧❛t✐♦♥s✮ ❲r✐t❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t② p ❛s ❛ ♣r♦❞✉❝t ♦❢ ❧❡ss s♠❛❧❧ ♣r♦❜❛❜✐❧✐t✐❡s ❛♥❞ ❡st✐♠❛t❡ t❤❡♠ ✇✐t❤ ▼❈▼❈✿ ❧❡t ( q i ) i =0 ..m ❜❡ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ✇✐t❤ q 0 = −∞ ❛♥❞ q m = q ✿ p = P [ g ( X ) > q ] = P [ g ( X ) > q m | g ( X ) > q m − 1 ] × P [ g ( X ) > q m − 1 ] = P [ g ( X ) > q ] = P [ g ( X ) > q m | g ( X ) > q m − 1 ] × · · · × P [ g ( X ) > q 1 ] ⇒ ❤♦✇ t♦ ❝❤♦♦s❡ ( q i ) i ✿ ❜❡❢♦r❡❤❛♥❞ ♦r ♦♥✲t❤❡✲❣♦❄ ❖♣t✐♠❛❧✐t②❄ ❇✐❛s❄ ❙é♠✐♥❛✐r❡ ❙ 3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✴✷✻
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