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❘❛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✐♦♥ µX g ❛ ✧❜❧❛❝❦✲❜♦①✧ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥t✐♥❣ ❛ ❝♦♠♣✉t❡r ❝♦❞❡✿ g : Rd → R Y = g(X) t❤❡ r❡❛❧✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇❤✐❝❤ ❞❡s❝r✐❜❡s t❤❡ st❛t❡ ♦❢ t❤❡ s②st❡♠❀ ✐ts ❞✐str✐❜✉t✐♦♥ µY ✐s ✉♥❦♥♦✇♥

❯♥❝❡rt❛✐♥t② ◗✉❛♥t✐✜❝❛t✐♦♥✿ F = {x ∈ Rd | g(x) > q}

✜♥❞ p = P [X ∈ F] = µX(F) ❢♦r ❛ ❣✐✈❡♥ q ✜♥❞ q ❢♦r ❛ ❣✐✈❡♥ p

■ss✉❡s

p = µX(F) = µY ([q; +∞[) ≪ 1 ♥❡❡❞s t♦ ✉s❡ g t♦ ❣❡t F ♦r µY ✇❤✐❝❤ ✐s t✐♠❡ ❝♦st❧② ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t♦r ❤❛s ❛ ❈❱ δ2 ≈ 1/Np ⇒ N ≫ 1/p

❙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 ❛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❄

❙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 ❛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❄

❙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 ❛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❄

❙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 ❛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❄

❙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 (qi)i=0..m ❜❡ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ✇✐t❤ q0 = −∞ ❛♥❞ qm = q✿ p = P [g(X) > q] = P [g(X) > qm | g(X) > qm−1] × P [g(X) > qm−1] = P [g(X) > q] = P [g(X) > qm | g(X) > qm−1] × · · · × P [g(X) > q1] ❤♦✇ t♦ ❝❤♦♦s❡ ✿ ❜❡❢♦r❡❤❛♥❞ ♦r ♦♥✲t❤❡✲❣♦❄ ❖♣t✐♠❛❧✐t②❄ ❇✐❛s❄

❙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 (qi)i=0..m ❜❡ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ✇✐t❤ q0 = −∞ ❛♥❞ qm = q✿ p = P [g(X) > q] = P [g(X) > qm | g(X) > qm−1] × P [g(X) > qm−1] = P [g(X) > q] = P [g(X) > qm | g(X) > qm−1] × · · · × P [g(X) > q1] ⇒ ❤♦✇ t♦ ❝❤♦♦s❡ (qi)i✿ ❜❡❢♦r❡❤❛♥❞ ♦r ♦♥✲t❤❡✲❣♦❄ ❖♣t✐♠❛❧✐t②❄ ❇✐❛s❄

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✴✷✻

■♥tr♦❞✉❝t✐♦♥

❆ t②♣✐❝❛❧ ▼❙ ❛❧❣♦r✐t❤♠ ✇♦r❦s ❛s ❢♦❧❧♦✇s✿

✶ ❙❛♠♣❧❡ ❛ ▼♦♥t❡✲❈❛r❧♦ ♣♦♣✉❧❛t✐♦♥ (Xi)i ♦❢ s✐③❡ N❀

y = (g(X1), · · · , g(XN))❀ j = 0

✷ ❊st✐♠❛t❡ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② P [g(X) > qj+1 | g(X) > qj] ✸ ❘❡s❛♠♣❧❡ t❤❡ (Xi)i s✉❝❤ t❤❛t g(Xi) ≤ qj+1 ❝♦♥❞✐t✐♦♥❛❧❧② t♦ ❜❡

❣r❡❛t❡r t❤❛♥ qj+1 ✭t❤❡ ♦t❤❡r ♦♥❡s ❞♦♥✬t ❝❤❛♥❣❡✮

✹ j ← j + 1 ❛♥❞ r❡♣❡❛t ✉♥t✐❧ j = m

P❛r❛❧❧❡❧ ❝♦♠♣✉t❛t✐♦♥ ❛t ❡❛❝❤ ✐t❡r❛t✐♦♥ ✐♥ t❤❡ r❡s❛♠♣❧✐♥❣ st❡♣ ▼✐♥✐♠❛❧ ✈❛r✐❛♥❝❡ ✇❤❡♥ ❛❧❧ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ❛r❡ ❡q✉❛❧ ❬✹❪ ❆❞❛♣t✐✈❡ ▼✉❧t✐❧❡✈❡❧ ❙♣❧✐tt✐♥❣✿ ♦r

❡♠♣✐r✐❝❛❧ q✉❛♥t✐❧❡s ♦❢ ♦r❞❡r ❜✐❛s ❬✹✱ ✶❪❀ t❤❡ ♥✉♠❜❡r ♦❢ s✉❜s❡ts ❝♦♥✈❡r❣❡s t♦✇❛r❞ ❛ ❝♦♥st❛♥t ❡♠♣✐r✐❝❛❧ q✉❛♥t✐❧❡s ♦❢ ♦r❞❡r ♥♦ ❜✐❛s ❛♥❞ ❈▲❚ ❬✸✱ ✷❪ ♠✐♥✐♠❛❧ ✈❛r✐❛♥❝❡ ✇✐t❤ ✭▲❛st P❛rt✐❝❧❡ ❆❧❣♦r✐t❤♠ ❬✺✱ ✻❪✮❀ t❤❡ ♥✉♠❜❡r ♦❢ s✉❜s❡ts ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❧❛✇ ✇✐t❤ ♣❛r❛♠❡t❡r ❞✐s❛❜❧❡s ♣❛r❛❧❧❡❧ ❝♦♠♣✉t❛t✐♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✸✴✷✻

■♥tr♦❞✉❝t✐♦♥

❆ t②♣✐❝❛❧ ▼❙ ❛❧❣♦r✐t❤♠ ✇♦r❦s ❛s ❢♦❧❧♦✇s✿

✶ ❙❛♠♣❧❡ ❛ ▼♦♥t❡✲❈❛r❧♦ ♣♦♣✉❧❛t✐♦♥ (Xi)i ♦❢ s✐③❡ N❀

y = (g(X1), · · · , g(XN))❀ j = 0

✷ ❊st✐♠❛t❡ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② P [g(X) > qj+1 | g(X) > qj] ✸ ❘❡s❛♠♣❧❡ t❤❡ (Xi)i s✉❝❤ t❤❛t g(Xi) ≤ qj+1 ❝♦♥❞✐t✐♦♥❛❧❧② t♦ ❜❡

❣r❡❛t❡r t❤❛♥ qj+1 ✭t❤❡ ♦t❤❡r ♦♥❡s ❞♦♥✬t ❝❤❛♥❣❡✮

✹ j ← j + 1 ❛♥❞ r❡♣❡❛t ✉♥t✐❧ j = m

⇒ P❛r❛❧❧❡❧ ❝♦♠♣✉t❛t✐♦♥ ❛t ❡❛❝❤ ✐t❡r❛t✐♦♥ ✐♥ t❤❡ r❡s❛♠♣❧✐♥❣ st❡♣ ▼✐♥✐♠❛❧ ✈❛r✐❛♥❝❡ ✇❤❡♥ ❛❧❧ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ❛r❡ ❡q✉❛❧ ❬✹❪ ❆❞❛♣t✐✈❡ ▼✉❧t✐❧❡✈❡❧ ❙♣❧✐tt✐♥❣✿ ♦r

❡♠♣✐r✐❝❛❧ q✉❛♥t✐❧❡s ♦❢ ♦r❞❡r ❜✐❛s ❬✹✱ ✶❪❀ t❤❡ ♥✉♠❜❡r ♦❢ s✉❜s❡ts ❝♦♥✈❡r❣❡s t♦✇❛r❞ ❛ ❝♦♥st❛♥t ❡♠♣✐r✐❝❛❧ q✉❛♥t✐❧❡s ♦❢ ♦r❞❡r ♥♦ ❜✐❛s ❛♥❞ ❈▲❚ ❬✸✱ ✷❪ ♠✐♥✐♠❛❧ ✈❛r✐❛♥❝❡ ✇✐t❤ ✭▲❛st P❛rt✐❝❧❡ ❆❧❣♦r✐t❤♠ ❬✺✱ ✻❪✮❀ t❤❡ ♥✉♠❜❡r ♦❢ s✉❜s❡ts ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❧❛✇ ✇✐t❤ ♣❛r❛♠❡t❡r ❞✐s❛❜❧❡s ♣❛r❛❧❧❡❧ ❝♦♠♣✉t❛t✐♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✸✴✷✻

■♥tr♦❞✉❝t✐♦♥

❆ t②♣✐❝❛❧ ▼❙ ❛❧❣♦r✐t❤♠ ✇♦r❦s ❛s ❢♦❧❧♦✇s✿

✶ ❙❛♠♣❧❡ ❛ ▼♦♥t❡✲❈❛r❧♦ ♣♦♣✉❧❛t✐♦♥ (Xi)i ♦❢ s✐③❡ N❀

y = (g(X1), · · · , g(XN))❀ j = 0

✷ ❊st✐♠❛t❡ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② P [g(X) > qj+1 | g(X) > qj] ✸ ❘❡s❛♠♣❧❡ t❤❡ (Xi)i s✉❝❤ t❤❛t g(Xi) ≤ qj+1 ❝♦♥❞✐t✐♦♥❛❧❧② t♦ ❜❡

❣r❡❛t❡r t❤❛♥ qj+1 ✭t❤❡ ♦t❤❡r ♦♥❡s ❞♦♥✬t ❝❤❛♥❣❡✮

✹ j ← j + 1 ❛♥❞ r❡♣❡❛t ✉♥t✐❧ j = m

⇒ P❛r❛❧❧❡❧ ❝♦♠♣✉t❛t✐♦♥ ❛t ❡❛❝❤ ✐t❡r❛t✐♦♥ ✐♥ t❤❡ r❡s❛♠♣❧✐♥❣ st❡♣ ▼✐♥✐♠❛❧ ✈❛r✐❛♥❝❡ ✇❤❡♥ ❛❧❧ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ❛r❡ ❡q✉❛❧ ❬✹❪ ❆❞❛♣t✐✈❡ ▼✉❧t✐❧❡✈❡❧ ❙♣❧✐tt✐♥❣✿ qj+1 ← y(p0N) ♦r qj+1 ← y(k)

❡♠♣✐r✐❝❛❧ q✉❛♥t✐❧❡s ♦❢ ♦r❞❡r p0 ∈ (0, 1) ⇒ ❜✐❛s ❬✹✱ ✶❪❀ t❤❡ ♥✉♠❜❡r ♦❢ s✉❜s❡ts ❝♦♥✈❡r❣❡s t♦✇❛r❞ ❛ ❝♦♥st❛♥t log p/ log p0 ❡♠♣✐r✐❝❛❧ q✉❛♥t✐❧❡s ♦❢ ♦r❞❡r k/N ⇒ ♥♦ ❜✐❛s ❛♥❞ ❈▲❚ ❬✸✱ ✷❪ ♠✐♥✐♠❛❧ ✈❛r✐❛♥❝❡ ✇✐t❤ k = 1 ✭▲❛st P❛rt✐❝❧❡ ❆❧❣♦r✐t❤♠ ❬✺✱ ✻❪✮❀ t❤❡ ♥✉♠❜❡r ♦❢ s✉❜s❡ts ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❧❛✇ ✇✐t❤ ♣❛r❛♠❡t❡r −N log p ⇒ ❞✐s❛❜❧❡s ♣❛r❛❧❧❡❧ ❝♦♠♣✉t❛t✐♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✸✴✷✻

❖✉t❧✐♥❡

✶ ■♥❝r❡❛s✐♥❣ r❛♥❞♦♠ ✇❛❧❦ ✷ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✸ ◗✉❛♥t✐❧❡ ❡st✐♠❛t✐♦♥ ✹ ❉❡s✐❣♥ ♣♦✐♥ts ✺ ❈♦♥❝❧✉s✐♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✹✴✷✻

❖✉t❧✐♥❡

✶ ■♥❝r❡❛s✐♥❣ r❛♥❞♦♠ ✇❛❧❦ ✷ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✸ ◗✉❛♥t✐❧❡ ❡st✐♠❛t✐♦♥ ✹ ❉❡s✐❣♥ ♣♦✐♥ts ✺ ❈♦♥❝❧✉s✐♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✺✴✷✻

■♥❝r❡❛s✐♥❣ r❛♥❞♦♠ ✇❛❧❦

❉❡✜♥✐t✐♦♥

❉❡✜♥✐t✐♦♥

▲❡t Y ❜❡ ❛ r❡❛❧✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ❞✐str✐❜✉t✐♦♥ µY ❛♥❞ ❝❞❢ F ✭❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s✮✳ ❖♥❡ ❝♦♥s✐❞❡rs t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ✭✇✐t❤ Y0 = −∞✮ s✉❝❤ t❤❛t✿ ∀n ∈ N : P [Yn+1 ∈ A | Y0, · · · , Yn] = µY (A ∩ (Yn, +∞)) µY ((Yn, +∞)) ✐✳❡✳ Yn+1 ✐s r❛♥❞♦♠❧② ❣r❡❛t❡r t❤❛♥ Yn✿ Yn+1 ∼ µY (· | Y > Yn) t❤❡ ❛r❡ ❞✐str✐❜✉t❡❞ ❛s t❤❡ ❛rr✐✈❛❧ t✐♠❡s ♦❢ ❛ P♦✐ss♦♥ ♣r♦❝❡ss ✇✐t❤ ♣❛r❛♠❡t❡r ✶ ❬✺✱ ✼❪ ❚✐♠❡ ✐♥ t❤❡ P♦✐ss♦♥ ♣r♦❝❡ss ✐s ❧✐♥❦❡❞ ✇✐t❤ r❛r✐t② ✐♥ ♣r♦❜❛❜✐❧✐t② ❚❤❡ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts ❜❡❢♦r❡ ✐s r❡❧❛t❡❞ t♦

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✻✴✷✻

■♥❝r❡❛s✐♥❣ r❛♥❞♦♠ ✇❛❧❦

❉❡✜♥✐t✐♦♥

❉❡✜♥✐t✐♦♥

▲❡t Y ❜❡ ❛ r❡❛❧✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ❞✐str✐❜✉t✐♦♥ µY ❛♥❞ ❝❞❢ F ✭❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s✮✳ ❖♥❡ ❝♦♥s✐❞❡rs t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ✭✇✐t❤ Y0 = −∞✮ s✉❝❤ t❤❛t✿ ∀n ∈ N : P [Yn+1 ∈ A | Y0, · · · , Yn] = µY (A ∩ (Yn, +∞)) µY ((Yn, +∞)) ✐✳❡✳ Yn+1 ✐s r❛♥❞♦♠❧② ❣r❡❛t❡r t❤❛♥ Yn✿ Yn+1 ∼ µY (· | Y > Yn) t❤❡ Tn = − log(P(Y > Yn)) ❛r❡ ❞✐str✐❜✉t❡❞ ❛s t❤❡ ❛rr✐✈❛❧ t✐♠❡s ♦❢ ❛ P♦✐ss♦♥ ♣r♦❝❡ss ✇✐t❤ ♣❛r❛♠❡t❡r ✶ ❬✺✱ ✼❪ ❚✐♠❡ ✐♥ t❤❡ P♦✐ss♦♥ ♣r♦❝❡ss ✐s ❧✐♥❦❡❞ ✇✐t❤ r❛r✐t② ✐♥ ♣r♦❜❛❜✐❧✐t② ❚❤❡ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts ❜❡❢♦r❡ ✐s r❡❧❛t❡❞ t♦

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✻✴✷✻

■♥❝r❡❛s✐♥❣ r❛♥❞♦♠ ✇❛❧❦

❉❡✜♥✐t✐♦♥

❉❡✜♥✐t✐♦♥

▲❡t Y ❜❡ ❛ r❡❛❧✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ❞✐str✐❜✉t✐♦♥ µY ❛♥❞ ❝❞❢ F ✭❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s✮✳ ❖♥❡ ❝♦♥s✐❞❡rs t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ✭✇✐t❤ Y0 = −∞✮ s✉❝❤ t❤❛t✿ ∀n ∈ N : P [Yn+1 ∈ A | Y0, · · · , Yn] = µY (A ∩ (Yn, +∞)) µY ((Yn, +∞)) ✐✳❡✳ Yn+1 ✐s r❛♥❞♦♠❧② ❣r❡❛t❡r t❤❛♥ Yn✿ Yn+1 ∼ µY (· | Y > Yn) t❤❡ Tn = − log(P(Y > Yn)) ❛r❡ ❞✐str✐❜✉t❡❞ ❛s t❤❡ ❛rr✐✈❛❧ t✐♠❡s ♦❢ ❛ P♦✐ss♦♥ ♣r♦❝❡ss ✇✐t❤ ♣❛r❛♠❡t❡r ✶ ❬✺✱ ✼❪ ❚✐♠❡ ✐♥ t❤❡ P♦✐ss♦♥ ♣r♦❝❡ss ✐s ❧✐♥❦❡❞ ✇✐t❤ r❛r✐t② ✐♥ ♣r♦❜❛❜✐❧✐t② ❚❤❡ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts ❜❡❢♦r❡ ✐s r❡❧❛t❡❞ t♦

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✻✴✷✻

■♥❝r❡❛s✐♥❣ r❛♥❞♦♠ ✇❛❧❦

❉❡✜♥✐t✐♦♥

❉❡✜♥✐t✐♦♥

▲❡t Y ❜❡ ❛ r❡❛❧✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ❞✐str✐❜✉t✐♦♥ µY ❛♥❞ ❝❞❢ F ✭❛ss✉♠❡❞ t♦ ❜❡ ❝♦♥t✐♥✉♦✉s✮✳ ❖♥❡ ❝♦♥s✐❞❡rs t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ✭✇✐t❤ Y0 = −∞✮ s✉❝❤ t❤❛t✿ ∀n ∈ N : P [Yn+1 ∈ A | Y0, · · · , Yn] = µY (A ∩ (Yn, +∞)) µY ((Yn, +∞)) ✐✳❡✳ Yn+1 ✐s r❛♥❞♦♠❧② ❣r❡❛t❡r t❤❛♥ Yn✿ Yn+1 ∼ µY (· | Y > Yn) t❤❡ Tn = − log(P(Y > Yn)) ❛r❡ ❞✐str✐❜✉t❡❞ ❛s t❤❡ ❛rr✐✈❛❧ t✐♠❡s ♦❢ ❛ P♦✐ss♦♥ ♣r♦❝❡ss ✇✐t❤ ♣❛r❛♠❡t❡r ✶ ❬✺✱ ✼❪ ❚✐♠❡ ✐♥ t❤❡ P♦✐ss♦♥ ♣r♦❝❡ss ✐s ❧✐♥❦❡❞ ✇✐t❤ r❛r✐t② ✐♥ ♣r♦❜❛❜✐❧✐t② ❚❤❡ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts My ❜❡❢♦r❡ y ∈ R ✐s r❡❧❛t❡❞ t♦ P [Y > y] = py My = Mt=− log py

L

∼ P(− log py)

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✻✴✷✻

■♥❝r❡❛s✐♥❣ r❛♥❞♦♠ ✇❛❧❦

❉❡✜♥✐t✐♦♥ ❋✐rst ❝♦♥s❡q✉❡♥❝❡✿ ♥✉♠❜❡r ♦❢ s✐♠✉❧❛t✐♦♥s t♦ ❣❡t t❤❡ r❡❛❧✐s❛t✐♦♥ ♦❢ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❛❜♦✈❡ ❛ ❣✐✈❡♥ t❤r❡s❤♦❧❞ ✭❡✈❡♥t ✇✐t❤ ♣r♦❜❛❜✐❧✐t② p✮ ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❧❛✇ P(log 1/p) ✐♥st❡❛❞ ♦❢ ❛ ●❡♦♠❡tr✐❝ ❧❛✇ G(p)✳ ✵ ✷✵ ✹✵ ✻✵ ✽✵ ✶✵✵ ✵✳✵✵ ✵✳✵✺ ✵✳✶✵ ✵✳✶✺ ✵✳✷✵ ◆ ❉❡♥s✐t②

❋✐❣✉r❡✿ ❈♦♠♣❛r✐s♦♥ ♦❢ P♦✐ss♦♥ ❛♥❞ ●❡♦♠❡tr✐❝ ❞❡♥s✐t✐❡s ✇✐t❤ p = 0.0228

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✼✴✷✻

■♥❝r❡❛s✐♥❣ r❛♥❞♦♠ ✇❛❧❦

❊①❛♠♣❧❡ Y ∼ N(0, 1) ❀ p = P [Y > 2] ≈ 2, 28.10−2 ❀ 1/p ≈ 43, 96 ❀ − log p ≈ 3, 78

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✽✴✷✻

✲✹ ✲✷ ✵ ✷ ✹ ✵✳✵ ✵✳✶ ✵✳✷ ✵✳✸ ✵✳✹ ① ❉❡♥s✐t②

P❧❛♥

✶ ■♥❝r❡❛s✐♥❣ r❛♥❞♦♠ ✇❛❧❦ ✷ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✸ ◗✉❛♥t✐❧❡ ❡st✐♠❛t✐♦♥ ✹ ❉❡s✐❣♥ ♣♦✐♥ts ✺ ❈♦♥❝❧✉s✐♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✾✴✷✻

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❡st✐♠❛t♦r

❈♦♥❝❡♣t

◆✉♠❜❡r ♦❢ ❡✈❡♥ts ❜❡❢♦r❡ t✐♠❡ t = − log(P(Y > q)) ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❧❛✇ ✇✐t❤ ♣❛r❛♠❡t❡r t ❡st✐♠❛t❡ t❤❡ ♣❛r❛♠❡t❡r ♦❢ ❛ P♦✐ss♦♥ ❧❛✇ ▲❡t ❜❡ ✐✐❞✳ ❘❱ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts ❛t t✐♠❡ ✿ ❀ ▲❛st P❛rt✐❝❧❡ ❊st✐♠❛t♦r✱ ❜✉t ✇✐t❤ ♣❛r❛❧❧❡❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❖♥❡ ❤❛s ✐♥❞❡❡❞ ❛♥ ❡st✐♠❛t♦r ♦❢ ✿

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✵✴✷✻

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❡st✐♠❛t♦r

❈♦♥❝❡♣t

◆✉♠❜❡r ♦❢ ❡✈❡♥ts ❜❡❢♦r❡ t✐♠❡ t = − log(P(Y > q)) ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❧❛✇ ✇✐t❤ ♣❛r❛♠❡t❡r t ❡st✐♠❛t❡ t❤❡ ♣❛r❛♠❡t❡r ♦❢ ❛ P♦✐ss♦♥ ❧❛✇ ▲❡t (Mi)i=1..N ❜❡ N ✐✐❞✳ ❘❱ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts ❛t t✐♠❡ t = − log p✿ Mi ∼ P(− log p)❀ Mq =

N

• i=1

Mi ∼ P(−N log p) ▲❛st P❛rt✐❝❧❡ ❊st✐♠❛t♦r✱ ❜✉t ✇✐t❤ ♣❛r❛❧❧❡❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❖♥❡ ❤❛s ✐♥❞❡❡❞ ❛♥ ❡st✐♠❛t♦r ♦❢ ✿

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✵✴✷✻

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❡st✐♠❛t♦r

❈♦♥❝❡♣t

◆✉♠❜❡r ♦❢ ❡✈❡♥ts ❜❡❢♦r❡ t✐♠❡ t = − log(P(Y > q)) ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❧❛✇ ✇✐t❤ ♣❛r❛♠❡t❡r t ❡st✐♠❛t❡ t❤❡ ♣❛r❛♠❡t❡r ♦❢ ❛ P♦✐ss♦♥ ❧❛✇ ▲❡t (Mi)i=1..N ❜❡ N ✐✐❞✳ ❘❱ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts ❛t t✐♠❡ t = − log p✿ Mi ∼ P(− log p)❀ Mq =

N

• i=1

Mi ∼ P(−N log p)

• − log p ≈ 1

N

N

• i=1

Mi = Mq N − → p =

• 1 − 1

N Mq ▲❛st P❛rt✐❝❧❡ ❊st✐♠❛t♦r✱ ❜✉t ✇✐t❤ ♣❛r❛❧❧❡❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❖♥❡ ❤❛s ✐♥❞❡❡❞ ❛♥ ❡st✐♠❛t♦r ♦❢ ✿

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✵✴✷✻

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❡st✐♠❛t♦r

❈♦♥❝❡♣t

◆✉♠❜❡r ♦❢ ❡✈❡♥ts ❜❡❢♦r❡ t✐♠❡ t = − log(P(Y > q)) ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❧❛✇ ✇✐t❤ ♣❛r❛♠❡t❡r t ❡st✐♠❛t❡ t❤❡ ♣❛r❛♠❡t❡r ♦❢ ❛ P♦✐ss♦♥ ❧❛✇ ▲❡t (Mi)i=1..N ❜❡ N ✐✐❞✳ ❘❱ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts ❛t t✐♠❡ t = − log p✿ Mi ∼ P(− log p)❀ Mq =

N

• i=1

Mi ∼ P(−N log p)

• − log p ≈ 1

N

N

• i=1

Mi = Mq N − → p =

• 1 − 1

N Mq ⇒ ▲❛st P❛rt✐❝❧❡ ❊st✐♠❛t♦r✱ ❜✉t ✇✐t❤ ♣❛r❛❧❧❡❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❖♥❡ ❤❛s ✐♥❞❡❡❞ ❛♥ ❡st✐♠❛t♦r ♦❢ ✿

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✵✴✷✻

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❡st✐♠❛t♦r

❈♦♥❝❡♣t

◆✉♠❜❡r ♦❢ ❡✈❡♥ts ❜❡❢♦r❡ t✐♠❡ t = − log(P(Y > q)) ❢♦❧❧♦✇s ❛ P♦✐ss♦♥ ❧❛✇ ✇✐t❤ ♣❛r❛♠❡t❡r t ❡st✐♠❛t❡ t❤❡ ♣❛r❛♠❡t❡r ♦❢ ❛ P♦✐ss♦♥ ❧❛✇ ▲❡t (Mi)i=1..N ❜❡ N ✐✐❞✳ ❘❱ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts ❛t t✐♠❡ t = − log p✿ Mi ∼ P(− log p)❀ Mq =

N

• i=1

Mi ∼ P(−N log p)

• − log p ≈ 1

N

N

• i=1

Mi = Mq N − → p =

• 1 − 1

N Mq ⇒ ▲❛st P❛rt✐❝❧❡ ❊st✐♠❛t♦r✱ ❜✉t ✇✐t❤ ♣❛r❛❧❧❡❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ⇒ ❖♥❡ ❤❛s ✐♥❞❡❡❞ ❛♥ ❡st✐♠❛t♦r ♦❢ P [Y > q0] , ∀q0 ≤ q✿ FN(y) = 1 −

• 1 − 1

N My

a.s.

− − − − →

N→∞ F(y)

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✵✴✷✻

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

❊①❛♠♣❧❡ Y ∼ N(0, 1) ❀ p = P [Y > 2] ≈ 2, 28.10−2 ❀ 1/p ≈ 43, 96 ❀ − log p ≈ 3, 78

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✶✴✷✻

✲✹ ✲✷ ✵ ✷ ✹ ✵✳✵ ✵✳✹ ✵✳✽ ② ❊♠♣✐r✐❝❛❧ ❝❞❢

◆ ❂ ✷ ❀ ♣ ❂ ✵✳✵✵✼✽

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

Pr❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ■❞❡❛❧❧②✱ ♦♥❡ ❦♥♦✇s ❤♦✇ t♦ ❣❡♥❡r❛t❡ t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ■♥ ♣r❛❝t✐❝❡✱ Y = g(X) ❛♥❞ ♦♥❡ ❝❛♥ ✉s❡ ▼❛r❦♦✈ ❝❤❛✐♥ s❛♠♣❧✐♥❣ ✭❡✳❣✳ ▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣s ♦r ●✐❜❜s ❛❧❣♦r✐t❤♠s✮ r❡q✉✐r❡s t♦ ✇♦r❦ ✇✐t❤ ❛ ♣♦♣✉❧❛t✐♦♥ t♦ ❣❡t st❛rt✐♥❣ ♣♦✐♥ts ❜❛t❝❤❡s ♦❢ r❛♥❞♦♠ ✇❛❧❦s ❛r❡ ❣❡♥❡r❛t❡❞ t♦❣❡t❤❡r

• ❡♥❡r❛t✐♥❣

r❛♥❞♦♠ ✇❛❧❦s ❘❡q✉✐r❡✿ ✱

• ❡♥❡r❛t❡

❝♦♣✐❡s ❛❝❝♦r❞✐♥❣ t♦ ❀ ❀ ✇❤✐❧❡ ❞♦

✸✿

❢♦r ✐ ✐♥ ❞♦

✻✿

• ❡♥❡r❛t❡

❀ ❡♥❞ ❢♦r

✾✿ ❡♥❞ ✇❤✐❧❡

❘❡t✉r♥ ✱ ✱

❡❛❝❤ s❛♠♣❧❡ ✐s r❡s❛♠♣❧❡❞ ❛❝❝♦r❞✐♥❣ t♦ ✐ts ♦✇♥ ❧❡✈❡❧

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✷✴✷✻

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

Pr❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ■❞❡❛❧❧②✱ ♦♥❡ ❦♥♦✇s ❤♦✇ t♦ ❣❡♥❡r❛t❡ t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ■♥ ♣r❛❝t✐❝❡✱ Y = g(X) ❛♥❞ ♦♥❡ ❝❛♥ ✉s❡ ▼❛r❦♦✈ ❝❤❛✐♥ s❛♠♣❧✐♥❣ ✭❡✳❣✳ ▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣s ♦r ●✐❜❜s ❛❧❣♦r✐t❤♠s✮ r❡q✉✐r❡s t♦ ✇♦r❦ ✇✐t❤ ❛ ♣♦♣✉❧❛t✐♦♥ t♦ ❣❡t st❛rt✐♥❣ ♣♦✐♥ts ⇒ ❜❛t❝❤❡s ♦❢ k r❛♥❞♦♠ ✇❛❧❦s ❛r❡ ❣❡♥❡r❛t❡❞ t♦❣❡t❤❡r

• ❡♥❡r❛t✐♥❣

r❛♥❞♦♠ ✇❛❧❦s ❘❡q✉✐r❡✿ ✱

• ❡♥❡r❛t❡

❝♦♣✐❡s ❛❝❝♦r❞✐♥❣ t♦ ❀ ❀ ✇❤✐❧❡ ❞♦

✸✿

❢♦r ✐ ✐♥ ❞♦

✻✿

• ❡♥❡r❛t❡

❀ ❡♥❞ ❢♦r

✾✿ ❡♥❞ ✇❤✐❧❡

❘❡t✉r♥ ✱ ✱

❡❛❝❤ s❛♠♣❧❡ ✐s r❡s❛♠♣❧❡❞ ❛❝❝♦r❞✐♥❣ t♦ ✐ts ♦✇♥ ❧❡✈❡❧

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✷✴✷✻

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

Pr❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ■❞❡❛❧❧②✱ ♦♥❡ ❦♥♦✇s ❤♦✇ t♦ ❣❡♥❡r❛t❡ t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ■♥ ♣r❛❝t✐❝❡✱ Y = g(X) ❛♥❞ ♦♥❡ ❝❛♥ ✉s❡ ▼❛r❦♦✈ ❝❤❛✐♥ s❛♠♣❧✐♥❣ ✭❡✳❣✳ ▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣s ♦r ●✐❜❜s ❛❧❣♦r✐t❤♠s✮ r❡q✉✐r❡s t♦ ✇♦r❦ ✇✐t❤ ❛ ♣♦♣✉❧❛t✐♦♥ t♦ ❣❡t st❛rt✐♥❣ ♣♦✐♥ts ⇒ ❜❛t❝❤❡s ♦❢ k r❛♥❞♦♠ ✇❛❧❦s ❛r❡ ❣❡♥❡r❛t❡❞ t♦❣❡t❤❡r

• ❡♥❡r❛t✐♥❣ k r❛♥❞♦♠ ✇❛❧❦s

❘❡q✉✐r❡✿ k✱ q

• ❡♥❡r❛t❡ k ❝♦♣✐❡s (Xi)i=1..k ❛❝❝♦r❞✐♥❣ t♦ µX❀ Y ← (g(X1), · · · , g(Xk))❀ M = (0, · · · , 0)

✇❤✐❧❡ min Y < q ❞♦

✸✿

ind ← which Y < q ❢♦r ✐ ✐♥ ind ❞♦ Mi = Mi + 1

✻✿

• ❡♥❡r❛t❡ X∗ ∼ µX(· | X > g(Xi))

Xi ← X∗❀ Yi = g(X∗) ❡♥❞ ❢♦r

✾✿ ❡♥❞ ✇❤✐❧❡

❘❡t✉r♥ M✱ (Xi)i=1..N✱ (Yi)i=1..N

❡❛❝❤ s❛♠♣❧❡ ✐s r❡s❛♠♣❧❡❞ ❛❝❝♦r❞✐♥❣ t♦ ✐ts ♦✇♥ ❧❡✈❡❧

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✷✴✷✻

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

Pr❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ■❞❡❛❧❧②✱ ♦♥❡ ❦♥♦✇s ❤♦✇ t♦ ❣❡♥❡r❛t❡ t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ■♥ ♣r❛❝t✐❝❡✱ Y = g(X) ❛♥❞ ♦♥❡ ❝❛♥ ✉s❡ ▼❛r❦♦✈ ❝❤❛✐♥ s❛♠♣❧✐♥❣ ✭❡✳❣✳ ▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣s ♦r ●✐❜❜s ❛❧❣♦r✐t❤♠s✮ r❡q✉✐r❡s t♦ ✇♦r❦ ✇✐t❤ ❛ ♣♦♣✉❧❛t✐♦♥ t♦ ❣❡t st❛rt✐♥❣ ♣♦✐♥ts ⇒ ❜❛t❝❤❡s ♦❢ k r❛♥❞♦♠ ✇❛❧❦s ❛r❡ ❣❡♥❡r❛t❡❞ t♦❣❡t❤❡r

• ❡♥❡r❛t✐♥❣ k r❛♥❞♦♠ ✇❛❧❦s

❘❡q✉✐r❡✿ k✱ q

• ❡♥❡r❛t❡ k ❝♦♣✐❡s (Xi)i=1..k ❛❝❝♦r❞✐♥❣ t♦ µX❀ Y ← (g(X1), · · · , g(Xk))❀ M = (0, · · · , 0)

✇❤✐❧❡ min Y < q ❞♦

✸✿

ind ← which Y < q ❢♦r ✐ ✐♥ ind ❞♦ Mi = Mi + 1

✻✿

• ❡♥❡r❛t❡ X∗ ∼ µX(· | X > g(Xi))

Xi ← X∗❀ Yi = g(X∗) ❡♥❞ ❢♦r

✾✿ ❡♥❞ ✇❤✐❧❡

❘❡t✉r♥ M✱ (Xi)i=1..N✱ (Yi)i=1..N

⇒ ❡❛❝❤ s❛♠♣❧❡ ✐s r❡s❛♠♣❧❡❞ ❛❝❝♦r❞✐♥❣ t♦ ✐ts ♦✇♥ ❧❡✈❡❧

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✷✴✷✻

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

Pr❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ✭❢♦r ❝♦♥❞✐t✐♦♥❛❧ s❛♠♣❧✐♥❣✮ ✐s ✐♥❝r❡❛s❡❞ ✇❤❡♥ t❤❡ st❛rt✐♥❣ ♣♦✐♥t ❛❧r❡❛❞② ❢♦❧❧♦✇s t❤❡ t❛r❣❡t❡❞ ❞✐str✐❜✉t✐♦♥❀ ✷ ♣♦ss✐❜✐❧✐t✐❡s✿ st♦r❡ ❡❛❝❤ st❛t❡ (Xi)i ❛♥❞ ✐ts ❝♦rr❡s♣♦♥❞✐♥❣ ❧❡✈❡❧ r❡✲❞r❛✇ ♦♥❧② t❤❡ s♠❛❧❧❡st Xi ✭⇒ ▲❛st P❛rt✐❝❧❡ ❆❧❣♦r✐t❤♠✮ ⇒ ▲P❆ ✐s ♦♥❧② ♦♥❡ ♣♦ss✐❜❧❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤✐s ❡st✐♠❛t♦r

❈♦♠♣✉t✐♥❣ t✐♠❡ ✇✐t❤ ▲P❆ ✐♠♣❧❡♠❡♥t❛t✐♦♥

▲❡t

♣❛r ❜❡ t❤❡ r❛♥❞♦♠ t✐♠❡ ♦❢ ❣❡♥❡r❛t✐♥❣

r❛♥❞♦♠ ✇❛❧❦s ❜② ❜❛t❝❤❡s ♦❢ s✐③❡ ✭ st❛♥❞✐♥❣ ❢♦r ❛ ♥✉♠❜❡r ♦❢ ❝♦r❡s✮ ✇✐t❤ ❜✉r♥✲✐♥

♣❛r

♦❢ ❘❱

♣❛r

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✸✴✷✻

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

Pr❛❝t✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ ✭❢♦r ❝♦♥❞✐t✐♦♥❛❧ s❛♠♣❧✐♥❣✮ ✐s ✐♥❝r❡❛s❡❞ ✇❤❡♥ t❤❡ st❛rt✐♥❣ ♣♦✐♥t ❛❧r❡❛❞② ❢♦❧❧♦✇s t❤❡ t❛r❣❡t❡❞ ❞✐str✐❜✉t✐♦♥❀ ✷ ♣♦ss✐❜✐❧✐t✐❡s✿ st♦r❡ ❡❛❝❤ st❛t❡ (Xi)i ❛♥❞ ✐ts ❝♦rr❡s♣♦♥❞✐♥❣ ❧❡✈❡❧ r❡✲❞r❛✇ ♦♥❧② t❤❡ s♠❛❧❧❡st Xi ✭⇒ ▲❛st P❛rt✐❝❧❡ ❆❧❣♦r✐t❤♠✮ ⇒ ▲P❆ ✐s ♦♥❧② ♦♥❡ ♣♦ss✐❜❧❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤✐s ❡st✐♠❛t♦r

❈♦♠♣✉t✐♥❣ t✐♠❡ ✇✐t❤ ▲P❆ ✐♠♣❧❡♠❡♥t❛t✐♦♥

▲❡t t♣❛r ❜❡ t❤❡ r❛♥❞♦♠ t✐♠❡ ♦❢ ❣❡♥❡r❛t✐♥❣ N r❛♥❞♦♠ ✇❛❧❦s ❜② ❜❛t❝❤❡s ♦❢ s✐③❡ k = N/nc ✭nc st❛♥❞✐♥❣ ❢♦r ❛ ♥✉♠❜❡r ♦❢ ❝♦r❡s✮ ✇✐t❤ ❜✉r♥✲✐♥ T t♣❛r = max ♦❢ nc ❘❱ ∼ P(−k log p) E [t♣❛r] = T(log p)2 ncδ2  1 +

• ncδ2

(log p)2

• 2 log nc +

1 T log 1/p  

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✸✴✷✻

Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥

❈♦♠♣❛r✐s♦♥ ▼❡❛♥ ❝♦♠♣✉t❡r t✐♠❡ ❛❣❛✐♥st ❝♦❡✣❝✐❡♥t ♦❢ ✈❛r✐❛t✐♦♥✿ t❤❡ ❝♦st ♦❢ ❛♥ ❛❧❣♦r✐t❤♠ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❣❡♥❡r❛t❡❞ s❛♠♣❧❡s ✐♥ ❛ r♦✇ ❜② ❛ ❝♦r❡✳ ❲❡ ❛ss✉♠❡ nc ≥ 1 ❝♦r❡s ❛♥❞ ❜✉r♥✲✐♥ = T ❢♦r ▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣s ❆❧❣♦r✐t❤♠ ❚✐♠❡ ❈♦❡❢✳ ♦❢ ✈❛r✳ δ2 ❚✐♠❡s ❱❙ δ ▼♦♥t❡ ❈❛r❧♦ N/nc 1/Np 1/pδ2 ❆▼❙ T log p

log p0 N(1−p0) nc log p log p0 1−p0 Np0 (1−p0)2 p0(log p0)2 T(log p)2 ncδ2

▲P❆ −TN log p − log p

N T(log p)2 δ2

❘❛♥❞♦♠ ✇❛❧❦ −T N

nc log p

− log p

N T(log p)2 ncδ2

❜❡st ❆▼❙ ✇❤❡♥ p0 → 1 ▲P❆ ❜r✐♥❣s t❤❡ t❤❡♦r❡t✐❝❛❧❧② ❜❡st ❆▼❙ ❜✉t ✐s ♥♦t ♣❛r❛❧❧❡❧ ❘❛♥❞♦♠ ✇❛❧❦ ❛❧❧♦✇s ❢♦r t❛❦✐♥❣ p0 → 1 ✇❤✐❧❡ ❦❡❡♣✐♥❣ t❤❡ ♣❛r❛❧❧❡❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✹✴✷✻

P❧❛♥

✶ ■♥❝r❡❛s✐♥❣ r❛♥❞♦♠ ✇❛❧❦ ✷ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✸ ◗✉❛♥t✐❧❡ ❡st✐♠❛t✐♦♥ ✹ ❉❡s✐❣♥ ♣♦✐♥ts ✺ ❈♦♥❝❧✉s✐♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✺✴✷✻

◗✉❛♥t✐❧❡ ❡st✐♠❛t✐♦♥

❉❡✜♥✐t✐♦♥

❈♦♥❝❡♣t

❆♣♣r♦①✐♠❛t❡ ❛ t✐♠❡ t = − log p ✇✐t❤ t✐♠❡s ♦❢ ❛ P♦✐ss♦♥ ♣r♦❝❡ss❀ t❤❡ ❤✐❣❤❡r t❤❡ r❛t❡✱ t❤❡ ❞❡♥s❡r t❤❡ ❞✐s❝r❡t✐s❛t✐♦♥ ♦❢ [0; +∞[ ❚❤❡ ❝❡♥t❡r ♦❢ t❤❡ ✐♥t❡r✈❛❧ ❝♦♥✈❡r❣❡s t♦✇❛r❞ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❝❡♥tr❡❞ ✐♥ ✇✐t❤ s②♠♠❡tr✐❝ ♣❞❢ ✇✐t❤ ❈▲❚✿ ❇♦✉♥❞s ♦♥ ❜✐❛s ♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✻✴✷✻

◗✉❛♥t✐❧❡ ❡st✐♠❛t✐♦♥

❉❡✜♥✐t✐♦♥

❈♦♥❝❡♣t

❆♣♣r♦①✐♠❛t❡ ❛ t✐♠❡ t = − log p ✇✐t❤ t✐♠❡s ♦❢ ❛ P♦✐ss♦♥ ♣r♦❝❡ss❀ t❤❡ ❤✐❣❤❡r t❤❡ r❛t❡✱ t❤❡ ❞❡♥s❡r t❤❡ ❞✐s❝r❡t✐s❛t✐♦♥ ♦❢ [0; +∞[ ❚❤❡ ❝❡♥t❡r ♦❢ t❤❡ ✐♥t❡r✈❛❧ [TMt; TMt+1] ❝♦♥✈❡r❣❡s t♦✇❛r❞ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❝❡♥tr❡❞ ✐♥ t ✇✐t❤ s②♠♠❡tr✐❝ ♣❞❢ ✇✐t❤ ❈▲❚✿ ❇♦✉♥❞s ♦♥ ❜✐❛s ♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✻✴✷✻

◗✉❛♥t✐❧❡ ❡st✐♠❛t✐♦♥

❉❡✜♥✐t✐♦♥

❈♦♥❝❡♣t

❆♣♣r♦①✐♠❛t❡ ❛ t✐♠❡ t = − log p ✇✐t❤ t✐♠❡s ♦❢ ❛ P♦✐ss♦♥ ♣r♦❝❡ss❀ t❤❡ ❤✐❣❤❡r t❤❡ r❛t❡✱ t❤❡ ❞❡♥s❡r t❤❡ ❞✐s❝r❡t✐s❛t✐♦♥ ♦❢ [0; +∞[ ❚❤❡ ❝❡♥t❡r ♦❢ t❤❡ ✐♥t❡r✈❛❧ [TMt; TMt+1] ❝♦♥✈❡r❣❡s t♦✇❛r❞ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❝❡♥tr❡❞ ✐♥ t ✇✐t❤ s②♠♠❡tr✐❝ ♣❞❢

• q = 1

2 (Ym + Ym+1) ✇✐t❤ m = ⌊E[Mq]⌋ = ⌊−N log p⌋ ❈▲❚✿ ❇♦✉♥❞s ♦♥ ❜✐❛s ♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✻✴✷✻

◗✉❛♥t✐❧❡ ❡st✐♠❛t✐♦♥

❉❡✜♥✐t✐♦♥

❈♦♥❝❡♣t

❆♣♣r♦①✐♠❛t❡ ❛ t✐♠❡ t = − log p ✇✐t❤ t✐♠❡s ♦❢ ❛ P♦✐ss♦♥ ♣r♦❝❡ss❀ t❤❡ ❤✐❣❤❡r t❤❡ r❛t❡✱ t❤❡ ❞❡♥s❡r t❤❡ ❞✐s❝r❡t✐s❛t✐♦♥ ♦❢ [0; +∞[ ❚❤❡ ❝❡♥t❡r ♦❢ t❤❡ ✐♥t❡r✈❛❧ [TMt; TMt+1] ❝♦♥✈❡r❣❡s t♦✇❛r❞ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❝❡♥tr❡❞ ✐♥ t ✇✐t❤ s②♠♠❡tr✐❝ ♣❞❢

• q = 1

2 (Ym + Ym+1) ✇✐t❤ m = ⌊E[Mq]⌋ = ⌊−N log p⌋ ❈▲❚✿ √ N ( q − q)

L

− →

m→∞ N

• 0, −p2 log p

f(q)2

• ❇♦✉♥❞s ♦♥ ❜✐❛s ♦♥ O(1/N)

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✻✴✷✻

◗✉❛♥t✐❧❡ ❡st✐♠❛t✐♦♥

❊①❛♠♣❧❡ Y ∼ N(0, 1) ❀ p = P [Y > 2] ≈ 2, 28.10−2 ❀ 1/p ≈ 43, 96 ❀ − log p ≈ 3, 78

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✼✴✷✻

✲✹ ✲✷ ✵ ✷ ✹ ✲✵✳✶✵ ✵✳✵✵ ✵✳✶✵ ②

◆ ❂ ✷ ❀ q ❂ ✶✳✸✹✽

P❧❛♥

✶ ■♥❝r❡❛s✐♥❣ r❛♥❞♦♠ ✇❛❧❦ ✷ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✸ ◗✉❛♥t✐❧❡ ❡st✐♠❛t✐♦♥ ✹ ❉❡s✐❣♥ ♣♦✐♥ts ✺ ❈♦♥❝❧✉s✐♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✽✴✷✻

❉❡s✐❣♥ ♣♦✐♥ts

❆❧❣♦r✐t❤♠ ◆♦ ♥❡❡❞ ❢♦r ❡①❛❝t s❛♠♣❧✐♥❣ ✐❢ t❤❡ ❣♦❛❧ ✐s ♦♥❧② t♦ ❣❡t ❢❛✐❧✐♥❣ s❛♠♣❧❡s ⇒ ✉s❡ ♦❢ ❛ ♠❡t❛♠♦❞❡❧ ❢♦r ❝♦♥❞✐t✐♦♥❛❧ s❛♠♣❧✐♥❣

• ❡tt✐♥❣ N❢❛✐❧ ❢❛✐❧✐♥❣ s❛♠♣❧❡s

❙❛♠♣❧❡ ❛ ♠✐♥✐♠❛❧✲s✐③❡❞ ❉♦❊ ▲❡❛r♥ ❛ ✜rst ♠❡t❛♠♦❞❡❧ ✇✐t❤ trend = failure

✸✿ ❢♦r N❢❛✐❧ t✐♠❡s ❞♦

⊲ ❙✐♠✉❧❛t❡ t❤❡ r❛♥❞♦♠ ✇❛❧❦s ♦♥❡ ❛❢t❡r t❤❡ ♦t❤❡r ❙❛♠♣❧❡ X1 ∼ µX❀ y1 = g(X1)❀ m = 1❀ tr❛✐♥ t❤❡ ♠❡t❛♠♦❞❡❧ ✇❤✐❧❡ ym < q ❞♦

✻✿

Xm+1 = Xm❀ ym+1 = ym ❢♦r T t✐♠❡s ❞♦ ⊲ Ps❡✉❞♦ ❜✉r♥✲✐♥ X∗ ∼ K(Xm+1, ·)❀ g(X∗) = y∗ ⊲ K ✐s ❛ ❦❡r♥❡❧ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥ s❛♠♣❧✐♥❣

✾✿

■❢ y∗ > ym+1✱ ym+1 = y∗ ❛♥❞ Xm+1 = X∗ ❡♥❞ ❢♦r ym+1 = g(Xm+1)❀ tr❛✐♥ t❤❡ ♠❡t❛♠♦❞❡❧

✶✷✿

■❢ ym+1 < ym✱ Xm+1 = Xm❀ ym+1 = ym❀ m = m + 1 ❡♥❞ ✇❤✐❧❡ ❡♥❞ ❢♦r

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✶✾✴✷✻

❉❡s✐❣♥ ♣♦✐♥ts

❊①❛♠♣❧❡ P❛r❛❜♦❧✐❝ ❧✐♠✐t✲st❛t❡ ❢✉♥❝t✐♦♥✿ g : x ∈ R2 − → 5 − x2 − 0.5(x1 − 0.1)2

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✵✴✷✻

−5 5 −5 5 x y

❉❡s✐❣♥ ♣♦✐♥ts

❊①❛♠♣❧❡ ❆ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❢♦✉r ❜r❛♥❝❤❡s s❡r✐❛❧ s②st❡♠✿ g : x ∈ R2 − → min

• 3 + (x1 − x2)2

10 − | x1 + x2 | √ 2 , 7 √ 2− | x1 − x2 |

• ❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✶✴✷✻

−5 5 −5 5 x y

P❧❛♥

✶ ■♥❝r❡❛s✐♥❣ r❛♥❞♦♠ ✇❛❧❦ ✷ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✸ ◗✉❛♥t✐❧❡ ❡st✐♠❛t✐♦♥ ✹ ❉❡s✐❣♥ ♣♦✐♥ts ✺ ❈♦♥❝❧✉s✐♦♥

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✷✴✷✻

❈♦♥❝❧✉s✐♦♥ ❈♦♥❝❧✉s✐♦♥

❖♥❡ ❝♦♥s✐❞❡rs ▼❛r❦♦✈ ❝❤❛✐♥s ✐♥st❡❛❞ ♦❢ s❛♠♣❧❡s ⇒ N ✐s ❛ ♥✉♠❜❡r ♦❢ ♣r♦❝❡ss❡s ▲❡ts ❞❡✜♥❡ ♣❛r❛❧❧❡❧ ❡st✐♠❛t♦rs ❢♦r ♣r♦❜❛❜✐❧✐t✐❡s ❛♥❞ q✉❛♥t✐❧❡s ✭❛♥❞ ♠♦♠❡♥ts ❬✽❪✮ ❚✇✐♥s ♦❢ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t♦rs ✇✐t❤ ❛ ✧log ❛ttr✐❜✉t❡✧✿ s✐♠✐❧❛r st❛t✐st✐❝❛❧ ♣r♦♣❡rt✐❡s ❜✉t ❛❞❞✐♥❣ ❛ log t♦ t❤❡ 1/p ❢❛❝t♦r✿ var [ p▼❈] ≈ p2 Np → var [ p] ≈ p2 log 1/p N var [ q▼❈] ≈ p2 Nf(q)2p → var [ q] ≈ p2 log 1/p Nf(q)2

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✸✴✷✻

❈♦♥❝❧✉s✐♦♥ P❡rs♣❡❝t✐✈❡s

❆❞❛♣t❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❡st✐♠❛t♦r ❢♦r ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠ ✭♠✐♥ ♦r ♠❛①✮ Pr♦❜❧❡♠ ♦❢ ❝♦♥❞✐t✐♦♥❛❧ s✐♠✉❧❛t✐♦♥s ✭▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣s✮ ❇❡st ✉s❡ ♦❢ ❛ ♠❡t❛♠♦❞❡❧ ❆❞❛♣t❛t✐♦♥ ❢♦r ❞✐s❝♦♥t✐♥✉♦✉s ❘❱

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✹✴✷✻

❇✐❜❧✐♦❣r❛♣❤② ■

❙✲❑ ❆✉ ❛♥❞ ❏ ▲ ❇❡❝❦✳ ❊st✐♠❛t✐♦♥ ♦❢ s♠❛❧❧ ❢❛✐❧✉r❡ ♣r♦❜❛❜✐❧✐t✐❡s ✐♥ ❤✐❣❤ ❞✐♠❡♥s✐♦♥s ❜② s✉❜s❡t s✐♠✉❧❛t✐♦♥✳ Pr♦❜❛❜✐❧✐st✐❝ ❊♥❣✐♥❡❡r✐♥❣ ▼❡❝❤❛♥✐❝s✱ ✶✻✭✹✮✿✷✻✸✕✷✼✼✱ ✷✵✵✶✳ ❈❤❛r❧❡s✲❊❞♦✉❛r❞ ❇ré❤✐❡r✱ ▲✉❞♦✈✐❝ ●♦✉❞❡♥❡❣❡✱ ❛♥❞ ▲♦✐❝ ❚✉❞❡❧❛✳ ❈❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠ ❢♦r ❛❞❛♣t❛t✐✈❡ ♠✉❧t✐❧❡✈❡❧ s♣❧✐tt✐♥❣ ❡st✐♠❛t♦rs ✐♥ ❛♥ ✐❞❡❛❧✐③❡❞ s❡tt✐♥❣✳ ❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✶✺✵✶✳✵✶✸✾✾✱ ✷✵✶✺✳ ❈❤❛r❧❡s✲❊❞♦✉❛r❞ ❇ré❤✐❡r✱ ❚♦♥② ▲❡❧✐❡✈r❡✱ ❛♥❞ ▼❛t❤✐❛s ❘♦✉ss❡t✳ ❆♥❛❧②s✐s ♦❢ ❛❞❛♣t✐✈❡ ♠✉❧t✐❧❡✈❡❧ s♣❧✐tt✐♥❣ ❛❧❣♦r✐t❤♠s ✐♥ ❛♥ ✐❞❡❛❧✐③❡❞ ❝❛s❡✳ ❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✶✹✵✺✳✶✸✺✷✱ ✷✵✶✹✳ ❋ ❈ér♦✉✱ P ❉❡❧ ▼♦r❛❧✱ ❚ ❋✉r♦♥✱ ❛♥❞ ❆ ●✉②❛❞❡r✳ ❙❡q✉❡♥t✐❛❧ ▼♦♥t❡ ❈❛r❧♦ ❢♦r r❛r❡ ❡✈❡♥t ❡st✐♠❛t✐♦♥✳ ❙t❛t✐st✐❝s ❛♥❞ ❈♦♠♣✉t✐♥❣✱ ✷✷✭✸✮✿✼✾✺✕✽✵✽✱ ✷✵✶✷✳

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✺✴✷✻

❇✐❜❧✐♦❣r❛♣❤② ■■

❆ ●✉②❛❞❡r✱ ◆ ❍❡♥❣❛rt♥❡r✱ ❛♥❞ ❊ ▼❛t③♥❡r✲▲ø❜❡r✳ ❙✐♠✉❧❛t✐♦♥ ❛♥❞ ❡st✐♠❛t✐♦♥ ♦❢ ❡①tr❡♠❡ q✉❛♥t✐❧❡s ❛♥❞ ❡①tr❡♠❡ ♣r♦❜❛❜✐❧✐t✐❡s✳ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ✫ ❖♣t✐♠✐③❛t✐♦♥✱ ✻✹✭✷✮✿✶✼✶✕✶✾✻✱ ✷✵✶✶✳ ❊r✐❝ ❙✐♠♦♥♥❡t✳ ❈♦♠❜✐♥❛t♦r✐❛❧ ❛♥❛❧②s✐s ♦❢ t❤❡ ❛❞❛♣t✐✈❡ ❧❛st ♣❛rt✐❝❧❡ ♠❡t❤♦❞✳ ❙t❛t✐st✐❝s ❛♥❞ ❈♦♠♣✉t✐♥❣✱ ♣❛❣❡s ✶✕✷✵✱ ✷✵✶✹✳ ❈❧❡♠❡♥t ❲❛❧t❡r✳ ▼♦✈✐♥❣ P❛rt✐❝❧❡s✿ ❛ ♣❛r❛❧❧❡❧ ♦♣t✐♠❛❧ ▼✉❧t✐❧❡✈❡❧ ❙♣❧✐tt✐♥❣ ♠❡t❤♦❞ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ q✉❛♥t✐❧❡s ❡st✐♠❛t✐♦♥ ❛♥❞ ♠❡t❛✲♠♦❞❡❧ ❜❛s❡❞ ❛❧❣♦r✐t❤♠s✳ ❚♦ ❛♣♣❡❛r ✐♥ ❙tr✉❝t✉r❛❧ ❙❛❢❡t②✱ ✷✵✶✹✳ ❈❧❡♠❡♥t ❲❛❧t❡r✳ P♦✐♥t ♣r♦❝❡ss✲❜❛s❡❞ ❡st✐♠❛t✐♦♥ ♦❢ ❦t❤✲♦r❞❡r ♠♦♠❡♥t✳ ❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✶✹✶✷✳✻✸✻✽✱ ✷✵✶✹✳

❙é♠✐♥❛✐r❡ ❙3 | ▼❛r❝❤ ✶✸t❤ ✷✵✶✺ | P❆●❊ ✷✻✴✷✻

▼❡r❝✐ ✦

❈♦♠♠✐ss❛r✐❛t à ❧✬é♥❡r❣✐❡ ❛t♦♠✐q✉❡ ❡t ❛✉① é♥❡r❣✐❡s ❛❧t❡r♥❛t✐✈❡s ❈❊❆✱ ❉❆▼✱ ❉■❋✱ ❋✲✾✶✷✾✼ ❆r♣❛❥♦♥✱ ❋r❛♥❝❡

➱t❛❜❧✐ss❡♠❡♥t ♣✉❜❧✐❝ à ❝❛r❛❝tèr❡ ✐♥❞✉str✐❡❧ ❡t ❝♦♠♠❡r❝✐❛❧ | ❘❈❙ P❛r✐s ❇ ✼✼✺ ✻✽✺ ✵✶✾

❈❊❆ ❉❆▼ ❉■❋