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t rts t t tr tr Prt ttst

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SLIDE 1

❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s✱ ❈❤❡♥✲❙t❡✐♥ ♠❡t❤♦❞ ❛♥❞ ❡①tr❡♠❡ ✈❛❧✉❡ t❤❡♦r②

P❛rt❤❛♥✐❧ ❘♦②✱ ■♥❞✐❛♥ ❙t❛t✐st✐❝❛❧ ■♥st✐t✉t❡ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❆♥✐s❤ ●❤♦s❤ ❛♥❞ ▼❛①✐♠ ❑✐rs❡❜♦♠ ❆♣r✐❧ ✷✺✱ ✷✵✶✾

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶ ✴ ✸✵

slide-2
SLIDE 2

❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s

✼ ✷✹ ✶ ✷✹ ✼ ✶ ✸

✸ ✼

✶ ✸

✶ ✼ ✸

✶ ✸

✶ ✷

✶ ✸

✸ ✷ ✸ ◆♦t❡ t❤❛t ✼ ✸ ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

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SLIDE 3

❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s

✼ ✷✹ = ✶ ✷✹/✼ ✶ ✸

✸ ✼

✶ ✸

✶ ✼ ✸

✶ ✸

✶ ✷

✶ ✸

✸ ✷ ✸ ◆♦t❡ t❤❛t ✼ ✸ ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

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SLIDE 4

❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s

✼ ✷✹ = ✶ ✷✹/✼ = ✶ ✸ + ✸

✼

✶ ✸

✶ ✼ ✸

✶ ✸

✶ ✷

✶ ✸

✸ ✷ ✸ ◆♦t❡ t❤❛t ✼ ✸ ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

slide-5
SLIDE 5

❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s

✼ ✷✹ = ✶ ✷✹/✼ = ✶ ✸ + ✸

✼

= ✶ ✸ +

✶ ✼/✸

✶ ✸

✶ ✷

✶ ✸

✸ ✷ ✸ ◆♦t❡ t❤❛t ✼ ✸ ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

slide-6
SLIDE 6

❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s

✼ ✷✹ = ✶ ✷✹/✼ = ✶ ✸ + ✸

✼

= ✶ ✸ +

✶ ✼/✸

= ✶ ✸ +

✶ ✷+ ✶

✸

✸ ✷ ✸ ◆♦t❡ t❤❛t ✼ ✸ ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

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SLIDE 7

❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s

✼ ✷✹ = ✶ ✷✹/✼ = ✶ ✸ + ✸

✼

= ✶ ✸ +

✶ ✼/✸

= ✶ ✸ +

✶ ✷+ ✶

✸

:= [✸, ✷, ✸] ◆♦t❡ t❤❛t ✼ ✸ ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

slide-8
SLIDE 8

❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s

✼ ✷✹ = ✶ ✷✹/✼ = ✶ ✸ + ✸

✼

= ✶ ✸ +

✶ ✼/✸

= ✶ ✸ +

✶ ✷+ ✶

✸

:= [✸, ✷, ✸] ◆♦t❡ t❤❛t ✼ > ✸ > ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ✵ ✶ ✭✇✐t❤ ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

slide-9
SLIDE 9

❘❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s

✼ ✷✹ = ✶ ✷✹/✼ = ✶ ✸ + ✸

✼

= ✶ ✸ +

✶ ✼/✸

= ✶ ✸ +

✶ ✷+ ✶

✸

:= [✸, ✷, ✸] ◆♦t❡ t❤❛t ✼ > ✸ > ✶✳ ❚❤❡r❡❢♦r❡ ❜② ❊✉❝❧✐❞❡❛♥ ❆❧❣♦r✐t❤♠✱ ❛♥② r❛t✐♦♥❛❧ ♥✉♠❜❡r ω = p/q ∈ (✵, ✶) ✭✇✐t❤ gcd(p, q) = ✶✮ ✇✐❧❧ ❤❛✈❡ ❛ t❡r♠✐♥❛t✐♥❣ ✭r❡❣✉❧❛r✮ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷ ✴ ✸✵

slide-10
SLIDE 10

❈♦♥✈❡rs❡❧② ✳ ✳ ✳

❲❤❡♥❡✈❡r A✶, A✷, A✸, A✹ ∈ N✱ [A✶, A✷, A✸, A✹] := ✶ A✶ +

✶ A✷+

✶ A✸+ ✶ A✹

∈ (✵, ✶) ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ ✱

✶ ✷

✭✇✐t❤

✶ ✷

✮ ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r ✐♥ ✵ ✶ ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✸ ✴ ✸✵

slide-11
SLIDE 11

❈♦♥✈❡rs❡❧② ✳ ✳ ✳

❲❤❡♥❡✈❡r A✶, A✷, A✸, A✹ ∈ N✱ [A✶, A✷, A✸, A✹] := ✶ A✶ +

✶ A✷+

✶ A✸+ ✶ A✹

∈ (✵, ✶) ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ n✱ ω = [A✶, A✷, . . . An] ✭✇✐t❤ A✶, A✷, . . . An ∈ N✮ ✐s ❛ r❛t✐♦♥❛❧ ♥✉♠❜❡r ✐♥ (✵, ✶)✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✸ ✴ ✸✵

slide-12
SLIDE 12

◆♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥

❚❤❡♦r❡♠

❆ ♥✉♠❜❡r ω ∈ (✵, ✶) ❤❛s ❛ ✉♥✐q✉❡ ♥♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ω = ✶ A✶ +

✶ A✷+

✶ A✸+···

=: [A✶, A✷, A✸, . . .] ✭✇✐t❤ ❡❛❝❤ Ai ∈ N✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ω / ∈ Q✳ ❋✉rt❤❡r♠♦r❡ ✐♥ t❤✐s ❝❛s❡✱ t❤❡ tr✉♥❝❛t❡

✶ ✷

❛s ✳ ❈❛♥♦♥✐❝❛❧ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥✿

✶ ✷

✳ ❊①❛♠♣❧❡s✿

✷✷ ✼ ❛♥❞ ✸✺✺ ✶✶✸✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✹ ✴ ✸✵

slide-13
SLIDE 13

◆♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥

❚❤❡♦r❡♠

❆ ♥✉♠❜❡r ω ∈ (✵, ✶) ❤❛s ❛ ✉♥✐q✉❡ ♥♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ω = ✶ A✶ +

✶ A✷+

✶ A✸+···

=: [A✶, A✷, A✸, . . .] ✭✇✐t❤ ❡❛❝❤ Ai ∈ N✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ω / ∈ Q✳ ❋✉rt❤❡r♠♦r❡ ✐♥ t❤✐s ❝❛s❡✱ t❤❡ nth tr✉♥❝❛t❡ [A✶, A✷, . . . An] → ω ❛s n → ∞✳ ❈❛♥♦♥✐❝❛❧ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥✿

✶ ✷

✳ ❊①❛♠♣❧❡s✿

✷✷ ✼ ❛♥❞ ✸✺✺ ✶✶✸✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✹ ✴ ✸✵

slide-14
SLIDE 14

◆♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥

❚❤❡♦r❡♠

❆ ♥✉♠❜❡r ω ∈ (✵, ✶) ❤❛s ❛ ✉♥✐q✉❡ ♥♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ω = ✶ A✶ +

✶ A✷+

✶ A✸+···

=: [A✶, A✷, A✸, . . .] ✭✇✐t❤ ❡❛❝❤ Ai ∈ N✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ω / ∈ Q✳ ❋✉rt❤❡r♠♦r❡ ✐♥ t❤✐s ❝❛s❡✱ t❤❡ nth tr✉♥❝❛t❡ [A✶, A✷, . . . An] → ω ❛s n → ∞✳ ❈❛♥♦♥✐❝❛❧ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥✿ ω ≈ [A✶, A✷, . . . An]✳ ❊①❛♠♣❧❡s✿

✷✷ ✼ ❛♥❞ ✸✺✺ ✶✶✸✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✹ ✴ ✸✵

slide-15
SLIDE 15

◆♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥

❚❤❡♦r❡♠

❆ ♥✉♠❜❡r ω ∈ (✵, ✶) ❤❛s ❛ ✉♥✐q✉❡ ♥♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ω = ✶ A✶ +

✶ A✷+

✶ A✸+···

=: [A✶, A✷, A✸, . . .] ✭✇✐t❤ ❡❛❝❤ Ai ∈ N✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ω / ∈ Q✳ ❋✉rt❤❡r♠♦r❡ ✐♥ t❤✐s ❝❛s❡✱ t❤❡ nth tr✉♥❝❛t❡ [A✶, A✷, . . . An] → ω ❛s n → ∞✳ ❈❛♥♦♥✐❝❛❧ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥✿ ω ≈ [A✶, A✷, . . . An]✳ ❊①❛♠♣❧❡s✿ π ≈ ✷✷

✼

❛♥❞

✸✺✺ ✶✶✸✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✹ ✴ ✸✵

slide-16
SLIDE 16

◆♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥

❚❤❡♦r❡♠

❆ ♥✉♠❜❡r ω ∈ (✵, ✶) ❤❛s ❛ ✉♥✐q✉❡ ♥♦♥✲t❡r♠✐♥❛t✐♥❣ ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ω = ✶ A✶ +

✶ A✷+

✶ A✸+···

=: [A✶, A✷, A✸, . . .] ✭✇✐t❤ ❡❛❝❤ Ai ∈ N✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ω / ∈ Q✳ ❋✉rt❤❡r♠♦r❡ ✐♥ t❤✐s ❝❛s❡✱ t❤❡ nth tr✉♥❝❛t❡ [A✶, A✷, . . . An] → ω ❛s n → ∞✳ ❈❛♥♦♥✐❝❛❧ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛t✐♦♥✿ ω ≈ [A✶, A✷, . . . An]✳ ❊①❛♠♣❧❡s✿ π ≈ ✷✷

✼ ❛♥❞ π ≈ ✸✺✺ ✶✶✸✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✹ ✴ ✸✵

slide-17
SLIDE 17

❲❤② ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s❄

❈♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥s ❛r❡ ✐♠♣♦rt❛♥t ✐♥ ❛❧❣❡❜r❛✱ ❛♥❛❧②s✐s✱ ❝♦♠❜✐♥❛t♦r✐❝s✱ ❡r❣♦❞✐❝ t❤❡♦r②✱ ❣❡♦♠❡tr②✱ ♥✉♠❜❡r t❤❡♦r②✱ ♣r♦❜❛❜✐❧✐t②✱ ❡t❝✳✳ ❙❡❡✱ ❢♦r ❡①❛♠♣❧❡✱ ❑❤✐♥t❝❤✐♥❡ ✭✶✾✻✹✮✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✺ ✴ ✸✵

slide-18
SLIDE 18

❋♦r ❛♥ ✐rr❛t✐♦♥❛❧ ω ∈ (✵, ✶)

ω = ✶ ✶/ω = ✶ [✶/ω] + {✶/ω} =: ✶ A✶(ω) + T(ω) = ✶ A✶(ω) +

✶ A✶(T(ω))+T ✷(ω)

=: ✶ A✶(ω) +

✶ A✷(ω)+T ✷(ω)

= · · ·

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✻ ✴ ✸✵

slide-19
SLIDE 19

❚❤❡ ●❛✉ss ♠❛♣

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✳ ❉❡✜♥❡ ❛♥❞

✶

❜② ✶ ✭●❛✉ss ♠❛♣✮ ❛♥❞

✶

✶ ❢♦r ✐rr❛t✐♦♥❛❧ ✭❛♥❞ ❛♥② ✇❛② ②♦✉ ❧✐❦❡ ❢♦r r❛t✐♦♥❛❧ ✮✳ ❋♦r ❛❧❧ ✱ s❡t

✶ ✶

❚❤❡♥ ❢♦r ❛❧♠♦st ❛❧❧ ✭♥❛♠❡❧②✱ ❢♦r ❛❧❧ ✮✱

✶ ✷ ✸

◗✉✐❝❦ ❖❜s❡r✈❛t✐♦♥✿ ✱

✶ ♠❡❛s✉r❛❜❧❡

❡❛❝❤ ♠❡❛s✉r❛❜❧❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✼ ✴ ✸✵

slide-20
SLIDE 20

❚❤❡ ●❛✉ss ♠❛♣

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✳ ❉❡✜♥❡ T : Ω → Ω ❛♥❞ A✶ : Ω → N ❜② T(ω) = {✶/ω} ✭●❛✉ss ♠❛♣✮ ❛♥❞ A✶(ω) = [✶/ω] ❢♦r ✐rr❛t✐♦♥❛❧ ω ∈ Ω ✭❛♥❞ ❛♥② ✇❛② ②♦✉ ❧✐❦❡ ❢♦r r❛t✐♦♥❛❧ ✮✳ ❋♦r ❛❧❧ ✱ s❡t

✶ ✶

❚❤❡♥ ❢♦r ❛❧♠♦st ❛❧❧ ✭♥❛♠❡❧②✱ ❢♦r ❛❧❧ ✮✱

✶ ✷ ✸

◗✉✐❝❦ ❖❜s❡r✈❛t✐♦♥✿ ✱

✶ ♠❡❛s✉r❛❜❧❡

❡❛❝❤ ♠❡❛s✉r❛❜❧❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✼ ✴ ✸✵

slide-21
SLIDE 21

❚❤❡ ●❛✉ss ♠❛♣

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✳ ❉❡✜♥❡ T : Ω → Ω ❛♥❞ A✶ : Ω → N ❜② T(ω) = {✶/ω} ✭●❛✉ss ♠❛♣✮ ❛♥❞ A✶(ω) = [✶/ω] ❢♦r ✐rr❛t✐♦♥❛❧ ω ∈ Ω ✭❛♥❞ ❛♥② ✇❛② ②♦✉ ❧✐❦❡ ❢♦r r❛t✐♦♥❛❧ ω ∈ Ω✮✳ ❋♦r ❛❧❧ ✱ s❡t

✶ ✶

❚❤❡♥ ❢♦r ❛❧♠♦st ❛❧❧ ✭♥❛♠❡❧②✱ ❢♦r ❛❧❧ ✮✱

✶ ✷ ✸

◗✉✐❝❦ ❖❜s❡r✈❛t✐♦♥✿ ✱

✶ ♠❡❛s✉r❛❜❧❡

❡❛❝❤ ♠❡❛s✉r❛❜❧❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✼ ✴ ✸✵

slide-22
SLIDE 22

❚❤❡ ●❛✉ss ♠❛♣

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✳ ❉❡✜♥❡ T : Ω → Ω ❛♥❞ A✶ : Ω → N ❜② T(ω) = {✶/ω} ✭●❛✉ss ♠❛♣✮ ❛♥❞ A✶(ω) = [✶/ω] ❢♦r ✐rr❛t✐♦♥❛❧ ω ∈ Ω ✭❛♥❞ ❛♥② ✇❛② ②♦✉ ❧✐❦❡ ❢♦r r❛t✐♦♥❛❧ ω ∈ Ω✮✳ ❋♦r ❛❧❧ j ∈ N✱ s❡t Aj+✶(ω) := A✶(T j(ω)), ω ∈ Ω. ❚❤❡♥ ❢♦r ❛❧♠♦st ❛❧❧ ✭♥❛♠❡❧②✱ ❢♦r ❛❧❧ ✮✱

✶ ✷ ✸

◗✉✐❝❦ ❖❜s❡r✈❛t✐♦♥✿ ✱

✶ ♠❡❛s✉r❛❜❧❡

❡❛❝❤ ♠❡❛s✉r❛❜❧❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✼ ✴ ✸✵

slide-23
SLIDE 23

❚❤❡ ●❛✉ss ♠❛♣

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✳ ❉❡✜♥❡ T : Ω → Ω ❛♥❞ A✶ : Ω → N ❜② T(ω) = {✶/ω} ✭●❛✉ss ♠❛♣✮ ❛♥❞ A✶(ω) = [✶/ω] ❢♦r ✐rr❛t✐♦♥❛❧ ω ∈ Ω ✭❛♥❞ ❛♥② ✇❛② ②♦✉ ❧✐❦❡ ❢♦r r❛t✐♦♥❛❧ ω ∈ Ω✮✳ ❋♦r ❛❧❧ j ∈ N✱ s❡t Aj+✶(ω) := A✶(T j(ω)), ω ∈ Ω. ❚❤❡♥ ❢♦r ❛❧♠♦st ❛❧❧ ω ∈ Ω ✭♥❛♠❡❧②✱ ❢♦r ❛❧❧ ω ∈ Ω \ Q✮✱ ω = [A✶(ω), A✷(ω), A✸(ω), . . .]. ◗✉✐❝❦ ❖❜s❡r✈❛t✐♦♥✿ ✱

✶ ♠❡❛s✉r❛❜❧❡

❡❛❝❤ ♠❡❛s✉r❛❜❧❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✼ ✴ ✸✵

slide-24
SLIDE 24

❚❤❡ ●❛✉ss ♠❛♣

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✳ ❉❡✜♥❡ T : Ω → Ω ❛♥❞ A✶ : Ω → N ❜② T(ω) = {✶/ω} ✭●❛✉ss ♠❛♣✮ ❛♥❞ A✶(ω) = [✶/ω] ❢♦r ✐rr❛t✐♦♥❛❧ ω ∈ Ω ✭❛♥❞ ❛♥② ✇❛② ②♦✉ ❧✐❦❡ ❢♦r r❛t✐♦♥❛❧ ω ∈ Ω✮✳ ❋♦r ❛❧❧ j ∈ N✱ s❡t Aj+✶(ω) := A✶(T j(ω)), ω ∈ Ω. ❚❤❡♥ ❢♦r ❛❧♠♦st ❛❧❧ ω ∈ Ω ✭♥❛♠❡❧②✱ ❢♦r ❛❧❧ ω ∈ Ω \ Q✮✱ ω = [A✶(ω), A✷(ω), A✸(ω), . . .]. ◗✉✐❝❦ ❖❜s❡r✈❛t✐♦♥✿ T✱ A✶ ♠❡❛s✉r❛❜❧❡ ⇒ ❡❛❝❤ An ♠❡❛s✉r❛❜❧❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✼ ✴ ✸✵

slide-25
SLIDE 25
  • ❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✳ ❉❡✜♥❡ T : Ω → Ω ❛♥❞ A✶ : Ω → N ❜② T(ω) = {✶/ω} ✭●❛✉ss ♠❛♣✮ ❛♥❞ A✶(ω) = [✶/ω]. ❇❛❞ ◆❡✇s✿ ❞♦❡s ♥♦t ♣r❡s❡r✈❡ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ ✵ ✶ ✳ ❉❡✜♥❡ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ✭●❛✉ss ♠❡❛s✉r❡✮ ♦♥ ❜② ✶ ✶ ✷

❚❤❡♦r❡♠ ✭●❛✉ss✮

♣r❡s❡r✈❡s ✱ ✐✳❡✳✱ ❢♦r ❛❧❧ ✱

✶

t❤❡ ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✽ ✴ ✸✵

slide-26
SLIDE 26
  • ❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✳ ❉❡✜♥❡ T : Ω → Ω ❛♥❞ A✶ : Ω → N ❜② T(ω) = {✶/ω} ✭●❛✉ss ♠❛♣✮ ❛♥❞ A✶(ω) = [✶/ω]. ❇❛❞ ◆❡✇s✿ T ❞♦❡s ♥♦t ♣r❡s❡r✈❡ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ (✵, ✶)✳ ❉❡✜♥❡ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ✭●❛✉ss ♠❡❛s✉r❡✮ ♦♥ ❜② ✶ ✶ ✷

❚❤❡♦r❡♠ ✭●❛✉ss✮

♣r❡s❡r✈❡s ✱ ✐✳❡✳✱ ❢♦r ❛❧❧ ✱

✶

t❤❡ ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✽ ✴ ✸✵

slide-27
SLIDE 27
  • ❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✳ ❉❡✜♥❡ T : Ω → Ω ❛♥❞ A✶ : Ω → N ❜② T(ω) = {✶/ω} ✭●❛✉ss ♠❛♣✮ ❛♥❞ A✶(ω) = [✶/ω]. ❇❛❞ ◆❡✇s✿ T ❞♦❡s ♥♦t ♣r❡s❡r✈❡ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ (✵, ✶)✳ ❉❡✜♥❡ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ P ✭●❛✉ss ♠❡❛s✉r❡✮ ♦♥ (Ω, A) ❜② P(A) =

  • A

✶ (✶ + x) log ✷dx.

❚❤❡♦r❡♠ ✭●❛✉ss✮

♣r❡s❡r✈❡s ✱ ✐✳❡✳✱ ❢♦r ❛❧❧ ✱

✶

t❤❡ ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✽ ✴ ✸✵

slide-28
SLIDE 28
  • ❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✳ ❉❡✜♥❡ T : Ω → Ω ❛♥❞ A✶ : Ω → N ❜② T(ω) = {✶/ω} ✭●❛✉ss ♠❛♣✮ ❛♥❞ A✶(ω) = [✶/ω]. ❇❛❞ ◆❡✇s✿ T ❞♦❡s ♥♦t ♣r❡s❡r✈❡ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ (✵, ✶)✳ ❉❡✜♥❡ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ P ✭●❛✉ss ♠❡❛s✉r❡✮ ♦♥ (Ω, A) ❜② P(A) =

  • A

✶ (✶ + x) log ✷dx.

❚❤❡♦r❡♠ ✭●❛✉ss✮

T ♣r❡s❡r✈❡s P✱ ✐✳❡✳✱ ❢♦r ❛❧❧ ✱

✶

t❤❡ ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✽ ✴ ✸✵

slide-29
SLIDE 29
  • ❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✳ ❉❡✜♥❡ T : Ω → Ω ❛♥❞ A✶ : Ω → N ❜② T(ω) = {✶/ω} ✭●❛✉ss ♠❛♣✮ ❛♥❞ A✶(ω) = [✶/ω]. ❇❛❞ ◆❡✇s✿ T ❞♦❡s ♥♦t ♣r❡s❡r✈❡ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ (✵, ✶)✳ ❉❡✜♥❡ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ P ✭●❛✉ss ♠❡❛s✉r❡✮ ♦♥ (Ω, A) ❜② P(A) =

  • A

✶ (✶ + x) log ✷dx.

❚❤❡♦r❡♠ ✭●❛✉ss✮

T ♣r❡s❡r✈❡s P✱ ✐✳❡✳✱ ❢♦r ❛❧❧ A ∈ A✱ P(A) = P(T −✶(A)). t❤❡ ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✽ ✴ ✸✵

slide-30
SLIDE 30
  • ❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✳ ❉❡✜♥❡ T : Ω → Ω ❛♥❞ A✶ : Ω → N ❜② T(ω) = {✶/ω} ✭●❛✉ss ♠❛♣✮ ❛♥❞ A✶(ω) = [✶/ω]. ❇❛❞ ◆❡✇s✿ T ❞♦❡s ♥♦t ♣r❡s❡r✈❡ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ♦♥ (✵, ✶)✳ ❉❡✜♥❡ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ P ✭●❛✉ss ♠❡❛s✉r❡✮ ♦♥ (Ω, A) ❜② P(A) =

  • A

✶ (✶ + x) log ✷dx.

❚❤❡♦r❡♠ ✭●❛✉ss✮

T ♣r❡s❡r✈❡s P✱ ✐✳❡✳✱ ❢♦r ❛❧❧ A ∈ A✱ P(A) = P(T −✶(A)). (Ω, A, P, T) = t❤❡ ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✽ ✴ ✸✵

slide-31
SLIDE 31

❆ r❡❢♦r♠✉❧❛t✐♦♥ ♦❢ ●❛✉ss✬s t❤❡♦r❡♠

❊①❡r❝✐s❡ ✭✐♥ Pr♦❜❛❜✐❧✐t② ❚❤❡♦r② ■■✮✿ ❙✉♣♣♦s❡ X ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❤❛✈✐♥❣ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ fX(x) = ✶ (✶ + x) log ✷, x ∈ (✵, ✶). ❚❤❡♥ s❤♦✇ t❤❛t {✶/X}

L

= X✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✾ ✴ ✸✵

slide-32
SLIDE 32

❆ st❛t✐♦♥❛r② ♣r♦❝❡ss

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✱ P(dx) = ((✶ + x) log ✷)−✶ dx✳ ❉❡✜♥❡ ❜② ✶ ❛♥❞

✶

❜②

✶

✶ ✳ ❋♦r ❛❧❧ ✱ s❡t

✶ ✶

❚❤✐s ❞❡✜♥❡s ❛ s❡q✉❡♥❝❡

✶ ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✲✈❛❧✉❡❞

r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦♥ t❤❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✳ ❞✐❣✐t ✐♥ t❤❡ r❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ ❛ r❛♥❞♦♠ ♥✉♠❜❡r ✵ ✶ ❝❤♦s❡♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛✇ ✳ ♣r❡s❡r✈❡s ✐s ❛ str✐❝t❧② st❛t✐♦♥❛r② ♣r♦❝❡ss✳ ■♥ ♣❛rt✐❝✉❧❛r✱

✶ ✷ ✸

❛r❡ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✵ ✴ ✸✵

slide-33
SLIDE 33

❆ st❛t✐♦♥❛r② ♣r♦❝❡ss

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✱ P(dx) = ((✶ + x) log ✷)−✶ dx✳ ❉❡✜♥❡ T : Ω → Ω ❜② T(ω) = {✶/ω} ❛♥❞ A✶ : Ω → N ❜② A✶(ω) = [✶/ω]✳ ❋♦r ❛❧❧ ✱ s❡t

✶ ✶

❚❤✐s ❞❡✜♥❡s ❛ s❡q✉❡♥❝❡

✶ ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✲✈❛❧✉❡❞

r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦♥ t❤❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✳ ❞✐❣✐t ✐♥ t❤❡ r❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ ❛ r❛♥❞♦♠ ♥✉♠❜❡r ✵ ✶ ❝❤♦s❡♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛✇ ✳ ♣r❡s❡r✈❡s ✐s ❛ str✐❝t❧② st❛t✐♦♥❛r② ♣r♦❝❡ss✳ ■♥ ♣❛rt✐❝✉❧❛r✱

✶ ✷ ✸

❛r❡ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✵ ✴ ✸✵

slide-34
SLIDE 34

❆ st❛t✐♦♥❛r② ♣r♦❝❡ss

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✱ P(dx) = ((✶ + x) log ✷)−✶ dx✳ ❉❡✜♥❡ T : Ω → Ω ❜② T(ω) = {✶/ω} ❛♥❞ A✶ : Ω → N ❜② A✶(ω) = [✶/ω]✳ ❋♦r ❛❧❧ j ∈ N✱ s❡t Aj+✶(ω) := A✶(T j(ω)), ω ∈ Ω. ❚❤✐s ❞❡✜♥❡s ❛ s❡q✉❡♥❝❡

✶ ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✲✈❛❧✉❡❞

r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦♥ t❤❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ✳ ❞✐❣✐t ✐♥ t❤❡ r❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ ❛ r❛♥❞♦♠ ♥✉♠❜❡r ✵ ✶ ❝❤♦s❡♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛✇ ✳ ♣r❡s❡r✈❡s ✐s ❛ str✐❝t❧② st❛t✐♦♥❛r② ♣r♦❝❡ss✳ ■♥ ♣❛rt✐❝✉❧❛r✱

✶ ✷ ✸

❛r❡ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✵ ✴ ✸✵

slide-35
SLIDE 35

❆ st❛t✐♦♥❛r② ♣r♦❝❡ss

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✱ P(dx) = ((✶ + x) log ✷)−✶ dx✳ ❉❡✜♥❡ T : Ω → Ω ❜② T(ω) = {✶/ω} ❛♥❞ A✶ : Ω → N ❜② A✶(ω) = [✶/ω]✳ ❋♦r ❛❧❧ j ∈ N✱ s❡t Aj+✶(ω) := A✶(T j(ω)), ω ∈ Ω. ❚❤✐s ❞❡✜♥❡s ❛ s❡q✉❡♥❝❡ {An : Ω → N}n≥✶ ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦♥ t❤❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω, A, P)✳ ❞✐❣✐t ✐♥ t❤❡ r❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ ❛ r❛♥❞♦♠ ♥✉♠❜❡r ✵ ✶ ❝❤♦s❡♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛✇ ✳ ♣r❡s❡r✈❡s ✐s ❛ str✐❝t❧② st❛t✐♦♥❛r② ♣r♦❝❡ss✳ ■♥ ♣❛rt✐❝✉❧❛r✱

✶ ✷ ✸

❛r❡ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✵ ✴ ✸✵

slide-36
SLIDE 36

❆ st❛t✐♦♥❛r② ♣r♦❝❡ss

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✱ P(dx) = ((✶ + x) log ✷)−✶ dx✳ ❉❡✜♥❡ T : Ω → Ω ❜② T(ω) = {✶/ω} ❛♥❞ A✶ : Ω → N ❜② A✶(ω) = [✶/ω]✳ ❋♦r ❛❧❧ j ∈ N✱ s❡t Aj+✶(ω) := A✶(T j(ω)), ω ∈ Ω. ❚❤✐s ❞❡✜♥❡s ❛ s❡q✉❡♥❝❡ {An : Ω → N}n≥✶ ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦♥ t❤❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω, A, P)✳ An = nth ❞✐❣✐t ✐♥ t❤❡ r❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ ❛ r❛♥❞♦♠ ♥✉♠❜❡r ω ∈ (✵, ✶) ❝❤♦s❡♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛✇ P✳ ♣r❡s❡r✈❡s ✐s ❛ str✐❝t❧② st❛t✐♦♥❛r② ♣r♦❝❡ss✳ ■♥ ♣❛rt✐❝✉❧❛r✱

✶ ✷ ✸

❛r❡ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✵ ✴ ✸✵

slide-37
SLIDE 37

❆ st❛t✐♦♥❛r② ♣r♦❝❡ss

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✱ P(dx) = ((✶ + x) log ✷)−✶ dx✳ ❉❡✜♥❡ T : Ω → Ω ❜② T(ω) = {✶/ω} ❛♥❞ A✶ : Ω → N ❜② A✶(ω) = [✶/ω]✳ ❋♦r ❛❧❧ j ∈ N✱ s❡t Aj+✶(ω) := A✶(T j(ω)), ω ∈ Ω. ❚❤✐s ❞❡✜♥❡s ❛ s❡q✉❡♥❝❡ {An : Ω → N}n≥✶ ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦♥ t❤❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω, A, P)✳ An = nth ❞✐❣✐t ✐♥ t❤❡ r❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ ❛ r❛♥❞♦♠ ♥✉♠❜❡r ω ∈ (✵, ✶) ❝❤♦s❡♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛✇ P✳ T ♣r❡s❡r✈❡s P ⇒ {An} ✐s ❛ str✐❝t❧② st❛t✐♦♥❛r② ♣r♦❝❡ss✳ ■♥ ♣❛rt✐❝✉❧❛r✱

✶ ✷ ✸

❛r❡ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✵ ✴ ✸✵

slide-38
SLIDE 38

❆ st❛t✐♦♥❛r② ♣r♦❝❡ss

❚❛❦❡ Ω = (✵, ✶)✱ A = B(✵,✶)✱ P(dx) = ((✶ + x) log ✷)−✶ dx✳ ❉❡✜♥❡ T : Ω → Ω ❜② T(ω) = {✶/ω} ❛♥❞ A✶ : Ω → N ❜② A✶(ω) = [✶/ω]✳ ❋♦r ❛❧❧ j ∈ N✱ s❡t Aj+✶(ω) := A✶(T j(ω)), ω ∈ Ω. ❚❤✐s ❞❡✜♥❡s ❛ s❡q✉❡♥❝❡ {An : Ω → N}n≥✶ ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✲✈❛❧✉❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦♥ t❤❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω, A, P)✳ An = nth ❞✐❣✐t ✐♥ t❤❡ r❡❣✉❧❛r ❝♦♥t✐♥✉❡❞ ❢r❛❝t✐♦♥ ❡①♣❛♥s✐♦♥ ♦❢ ❛ r❛♥❞♦♠ ♥✉♠❜❡r ω ∈ (✵, ✶) ❝❤♦s❡♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❧❛✇ P✳ T ♣r❡s❡r✈❡s P ⇒ {An} ✐s ❛ str✐❝t❧② st❛t✐♦♥❛r② ♣r♦❝❡ss✳ ■♥ ♣❛rt✐❝✉❧❛r✱ A✶, A✷, A✸, . . . ❛r❡ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✵ ✴ ✸✵

slide-39
SLIDE 39

❚✇♦ ❡❛s② ♦❜s❡r✈❛t✐♦♥s

❉✐r❡❝t ❈♦♠♣✉t❛t✐♦♥✿ ❋♦r ❛❧❧ m ∈ N✱ P(A✶ ≥ m) = ✶ log ✷ log

  • ✶ + ✶

m

  • ✶

✷ ❛s ❋♦r ❛❧❧ ✵✱

✶

✷

✶

✷ ✶ ❛s ✳ ■♥ ♣❛rt✐❝✉❧❛r✱

✶

✷

✶

✭

✶ ✐s r❡❣✉❧❛r❧② ✈❛r②✐♥❣ ✇✐t❤ ✐♥❞❡① ✶✮✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✶ ✴ ✸✵

slide-40
SLIDE 40

❚✇♦ ❡❛s② ♦❜s❡r✈❛t✐♦♥s

❉✐r❡❝t ❈♦♠♣✉t❛t✐♦♥✿ ❋♦r ❛❧❧ m ∈ N✱ P(A✶ ≥ m) = ✶ log ✷ log

  • ✶ + ✶

m

  • ∼

✶ m log ✷ (❛s m → ∞). ❋♦r ❛❧❧ ✵✱

✶

✷

✶

✷ ✶ ❛s ✳ ■♥ ♣❛rt✐❝✉❧❛r✱

✶

✷

✶

✭

✶ ✐s r❡❣✉❧❛r❧② ✈❛r②✐♥❣ ✇✐t❤ ✐♥❞❡① ✶✮✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✶ ✴ ✸✵

slide-41
SLIDE 41

❚✇♦ ❡❛s② ♦❜s❡r✈❛t✐♦♥s

❉✐r❡❝t ❈♦♠♣✉t❛t✐♦♥✿ ❋♦r ❛❧❧ m ∈ N✱ P(A✶ ≥ m) = ✶ log ✷ log

  • ✶ + ✶

m

  • ∼

✶ m log ✷ (❛s m → ∞). ❋♦r ❛❧❧ u > ✵✱ P A✶ log ✷ n > u

  • = P
  • A✶ ≥

un log ✷

  • ∼ ✶

un ❛s n → ∞✳ ■♥ ♣❛rt✐❝✉❧❛r✱

✶

✷

✶

✭

✶ ✐s r❡❣✉❧❛r❧② ✈❛r②✐♥❣ ✇✐t❤ ✐♥❞❡① ✶✮✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✶ ✴ ✸✵

slide-42
SLIDE 42

❚✇♦ ❡❛s② ♦❜s❡r✈❛t✐♦♥s

❉✐r❡❝t ❈♦♠♣✉t❛t✐♦♥✿ ❋♦r ❛❧❧ m ∈ N✱ P(A✶ ≥ m) = ✶ log ✷ log

  • ✶ + ✶

m

  • ∼

✶ m log ✷ (❛s m → ∞). ❋♦r ❛❧❧ u > ✵✱ P A✶ log ✷ n > u

  • = P
  • A✶ ≥

un log ✷

  • ∼ ✶

un ❛s n → ∞✳ ■♥ ♣❛rt✐❝✉❧❛r✱ nP A✶ log ✷ n > u

  • → u−✶

✭A✶ ✐s r❡❣✉❧❛r❧② ✈❛r②✐♥❣ ✇✐t❤ ✐♥❞❡① ✶✮✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✶ ✴ ✸✵

slide-43
SLIDE 43

■❢ A✶, A✷, A✸, . . . ✇❡r❡ ✐♥❞❡♣❡♥❞❡♥t

t❤❡♥ ✶(A✶ log ✷>un), ✶(A✷ log ✷>un), ✶(A✸ log ✷>un), . . .

iid

∼ Ber(pn), ✇❤❡r❡ pn = P(A✶ log ✷ > un)

✶

❚❤❡r❡❢♦r❡ ❢♦r ❛❧❧ ✵✱ ✶ ✷

✶

✶

✷ ✶

❛s ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✷ ✴ ✸✵

slide-44
SLIDE 44

■❢ A✶, A✷, A✸, . . . ✇❡r❡ ✐♥❞❡♣❡♥❞❡♥t

t❤❡♥ ✶(A✶ log ✷>un), ✶(A✷ log ✷>un), ✶(A✸ log ✷>un), . . .

iid

∼ Ber(pn), ✇❤❡r❡ pn = P(A✶ log ✷ > un) ∼

✶ un.

❚❤❡r❡❢♦r❡ ❢♦r ❛❧❧ ✵✱ ✶ ✷

✶

✶

✷ ✶

❛s ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✷ ✴ ✸✵

slide-45
SLIDE 45

■❢ A✶, A✷, A✸, . . . ✇❡r❡ ✐♥❞❡♣❡♥❞❡♥t

t❤❡♥ ✶(A✶ log ✷>un), ✶(A✷ log ✷>un), ✶(A✸ log ✷>un), . . .

iid

∼ Ber(pn), ✇❤❡r❡ pn = P(A✶ log ✷ > un) ∼

✶ un.

❚❤❡r❡❢♦r❡ ❢♦r ❛❧❧ u > ✵✱ Eu

n := #{✶ ≤ j ≤ n : Aj log ✷ > un} ✶

✶

✷ ✶

❛s ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✷ ✴ ✸✵

slide-46
SLIDE 46

■❢ A✶, A✷, A✸, . . . ✇❡r❡ ✐♥❞❡♣❡♥❞❡♥t

t❤❡♥ ✶(A✶ log ✷>un), ✶(A✷ log ✷>un), ✶(A✸ log ✷>un), . . .

iid

∼ Ber(pn), ✇❤❡r❡ pn = P(A✶ log ✷ > un) ∼

✶ un.

❚❤❡r❡❢♦r❡ ❢♦r ❛❧❧ u > ✵✱ Eu

n := #{✶ ≤ j ≤ n : Aj log ✷ > un} = n

  • j=✶

✶(Aj log ✷>un)

✶

❛s ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✷ ✴ ✸✵

slide-47
SLIDE 47

■❢ A✶, A✷, A✸, . . . ✇❡r❡ ✐♥❞❡♣❡♥❞❡♥t

t❤❡♥ ✶(A✶ log ✷>un), ✶(A✷ log ✷>un), ✶(A✸ log ✷>un), . . .

iid

∼ Ber(pn), ✇❤❡r❡ pn = P(A✶ log ✷ > un) ∼

✶ un.

❚❤❡r❡❢♦r❡ ❢♦r ❛❧❧ u > ✵✱ Eu

n := #{✶ ≤ j ≤ n : Aj log ✷ > un} = n

  • j=✶

✶(Aj log ✷>un) ∼ Bin(n, pn)

✶

❛s ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✷ ✴ ✸✵

slide-48
SLIDE 48

■❢ A✶, A✷, A✸, . . . ✇❡r❡ ✐♥❞❡♣❡♥❞❡♥t

t❤❡♥ ✶(A✶ log ✷>un), ✶(A✷ log ✷>un), ✶(A✸ log ✷>un), . . .

iid

∼ Ber(pn), ✇❤❡r❡ pn = P(A✶ log ✷ > un) ∼

✶ un.

❚❤❡r❡❢♦r❡ ❢♦r ❛❧❧ u > ✵✱ Eu

n := #{✶ ≤ j ≤ n : Aj log ✷ > un} = n

  • j=✶

✶(Aj log ✷>un) ∼ Bin(n, pn)

L

− → Eu

∞ ∼ Poi(u−✶)

❛s n → ∞✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✷ ✴ ✸✵

slide-49
SLIDE 49

❉♦❡❜❧✐♥✲■♦s✐❢❡s❝✉ ❛s②♠♣t♦t✐❝s

❚❤❡♦r❡♠ ✭❉♦❡❜❧✐♥ ✭✶✾✹✵✮✱ ■♦s✐❢❡s❝✉ ✭✶✾✼✼✮✮

❋♦r ❛❧❧ u > ✵✱ Eu

n := #{✶ ≤ j ≤ n : Aj log ✷ > un} L

− → Eu

∞ ∼ Poi(u−✶)

❛s n → ∞✳

❈♦r♦❧❧❛r② ✭▼❛✐♥ r❡s✉❧t ♦❢ ●❛❧❛♠❜♦s ✭✶✾✼✷✮✮

▲❡t

✶

✷ ✶ ✶ ✱ ✳ ❚❤❡♥ ❢♦r ❛❧❧ ✵✱

✶

✶

❛s ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✸ ✴ ✸✵

slide-50
SLIDE 50

❉♦❡❜❧✐♥✲■♦s✐❢❡s❝✉ ❛s②♠♣t♦t✐❝s

❚❤❡♦r❡♠ ✭❉♦❡❜❧✐♥ ✭✶✾✹✵✮✱ ■♦s✐❢❡s❝✉ ✭✶✾✼✼✮✮

❋♦r ❛❧❧ u > ✵✱ Eu

n := #{✶ ≤ j ≤ n : Aj log ✷ > un} L

− → Eu

∞ ∼ Poi(u−✶)

❛s n → ∞✳

❈♦r♦❧❧❛r② ✭▼❛✐♥ r❡s✉❧t ♦❢ ●❛❧❛♠❜♦s ✭✶✾✼✷✮✮

▲❡t M(✶)

n

:= max{Ai log ✷ : ✶ ≤ ✶ ≤ n}✱ n ∈ N✳ ❚❤❡♥ ❢♦r ❛❧❧ u > ✵✱ P

  • M(✶)

n

n ≤ u

  • → e−u−✶

❛s n → ∞✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✸ ✴ ✸✵

slide-51
SLIDE 51

❚❤❡ ♠❛✐♥ q✉❡st✐♦♥

❚❤❡♦r❡♠ ✭❉♦❡❜❧✐♥ ✭✶✾✹✵✮✱ ■♦s✐❢❡s❝✉ ✭✶✾✼✼✮✮

❋♦r ❛❧❧ u > ✵✱ (DI) Eu

n := #{✶ ≤ j ≤ n : Aj log ✷ > un} L

− → Eu

∞ ∼ Poi(u−✶)

❛s n → ∞✳

◗✉❡st✐♦♥

❲❤❛t ✐s t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ (DI)❄

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✹ ✴ ✸✵

slide-52
SLIDE 52

❲❤② ❞♦ ✇❡ ❝❛r❡❄

❈❛♥ ❡st✐♠❛t❡ t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❝❛❧❡❞ ♠❛①✐♠❛ s❡q✉❡♥❝❡

✶

✭❛s ✐♥ ●❛❧❛♠❜♦s ✭✶✾✼✷✮✮ ✲ s✐❣♥✐✜❝❛♥t❧② ✐♠♣r♦✈❡s ❛ r❡s✉❧t ♦❢ P❤✐❧✐♣♣ ✭✶✾✼✻✮✳ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ s❝❛❧❡❞ ♠❛①✐♠❛ ❢♦r ❛♥② ✭✉♥✐❢♦r♠ ♦✈❡r ✮✳ ❆ t✐♥② ❞❡t♦✉r ♦❢ ♦✉r ♣r♦♦❢ r❡❝♦✈❡rs ❛ r❡s✉❧t ♦❢ ❚②r❛♥✲❑❛♠✐➠s❦❛ ✭✷✵✶✵✮ ♦♥ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡①tr❡♠❛❧ ♣♦✐♥t ♣r♦❝❡ss✳ ✭■♥s♣✐r❡❞ ❜② ❈❤✐❛r✐♥✐✱ ❈✐♣r✐❛♥✐ ❛♥❞ ❍❛③r❛ ✭✷✵✶✺✮✳✮ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❝❛❧❡❞ ♠❛①✐♠❛ ❢♦r t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧❛r s✉r❢❛❝❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✺ ✴ ✸✵

slide-53
SLIDE 53

❲❤② ❞♦ ✇❡ ❝❛r❡❄

❈❛♥ ❡st✐♠❛t❡ t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❝❛❧❡❞ ♠❛①✐♠❛ s❡q✉❡♥❝❡ M(✶)

n /n ✭❛s ✐♥ ●❛❧❛♠❜♦s ✭✶✾✼✷✮✮

✲ s✐❣♥✐✜❝❛♥t❧② ✐♠♣r♦✈❡s ❛ r❡s✉❧t ♦❢ P❤✐❧✐♣♣ ✭✶✾✼✻✮✳ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ s❝❛❧❡❞ ♠❛①✐♠❛ ❢♦r ❛♥② ✭✉♥✐❢♦r♠ ♦✈❡r ✮✳ ❆ t✐♥② ❞❡t♦✉r ♦❢ ♦✉r ♣r♦♦❢ r❡❝♦✈❡rs ❛ r❡s✉❧t ♦❢ ❚②r❛♥✲❑❛♠✐➠s❦❛ ✭✷✵✶✵✮ ♦♥ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡①tr❡♠❛❧ ♣♦✐♥t ♣r♦❝❡ss✳ ✭■♥s♣✐r❡❞ ❜② ❈❤✐❛r✐♥✐✱ ❈✐♣r✐❛♥✐ ❛♥❞ ❍❛③r❛ ✭✷✵✶✺✮✳✮ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❝❛❧❡❞ ♠❛①✐♠❛ ❢♦r t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧❛r s✉r❢❛❝❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✺ ✴ ✸✵

slide-54
SLIDE 54

❲❤② ❞♦ ✇❡ ❝❛r❡❄

❈❛♥ ❡st✐♠❛t❡ t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❝❛❧❡❞ ♠❛①✐♠❛ s❡q✉❡♥❝❡ M(✶)

n /n ✭❛s ✐♥ ●❛❧❛♠❜♦s ✭✶✾✼✷✮✮ ✲ s✐❣♥✐✜❝❛♥t❧② ✐♠♣r♦✈❡s ❛ r❡s✉❧t

♦❢ P❤✐❧✐♣♣ ✭✶✾✼✻✮✳ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ s❝❛❧❡❞ ♠❛①✐♠❛ ❢♦r ❛♥② ✭✉♥✐❢♦r♠ ♦✈❡r ✮✳ ❆ t✐♥② ❞❡t♦✉r ♦❢ ♦✉r ♣r♦♦❢ r❡❝♦✈❡rs ❛ r❡s✉❧t ♦❢ ❚②r❛♥✲❑❛♠✐➠s❦❛ ✭✷✵✶✵✮ ♦♥ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡①tr❡♠❛❧ ♣♦✐♥t ♣r♦❝❡ss✳ ✭■♥s♣✐r❡❞ ❜② ❈❤✐❛r✐♥✐✱ ❈✐♣r✐❛♥✐ ❛♥❞ ❍❛③r❛ ✭✷✵✶✺✮✳✮ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❝❛❧❡❞ ♠❛①✐♠❛ ❢♦r t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧❛r s✉r❢❛❝❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✺ ✴ ✸✵

slide-55
SLIDE 55

❲❤② ❞♦ ✇❡ ❝❛r❡❄

❈❛♥ ❡st✐♠❛t❡ t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❝❛❧❡❞ ♠❛①✐♠❛ s❡q✉❡♥❝❡ M(✶)

n /n ✭❛s ✐♥ ●❛❧❛♠❜♦s ✭✶✾✼✷✮✮ ✲ s✐❣♥✐✜❝❛♥t❧② ✐♠♣r♦✈❡s ❛ r❡s✉❧t

♦❢ P❤✐❧✐♣♣ ✭✶✾✼✻✮✳ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ s❝❛❧❡❞ kth ♠❛①✐♠❛ ❢♦r ❛♥② k ∈ N ✭✉♥✐❢♦r♠ ♦✈❡r ✮✳ ❆ t✐♥② ❞❡t♦✉r ♦❢ ♦✉r ♣r♦♦❢ r❡❝♦✈❡rs ❛ r❡s✉❧t ♦❢ ❚②r❛♥✲❑❛♠✐➠s❦❛ ✭✷✵✶✵✮ ♦♥ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡①tr❡♠❛❧ ♣♦✐♥t ♣r♦❝❡ss✳ ✭■♥s♣✐r❡❞ ❜② ❈❤✐❛r✐♥✐✱ ❈✐♣r✐❛♥✐ ❛♥❞ ❍❛③r❛ ✭✷✵✶✺✮✳✮ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❝❛❧❡❞ ♠❛①✐♠❛ ❢♦r t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧❛r s✉r❢❛❝❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✺ ✴ ✸✵

slide-56
SLIDE 56

❲❤② ❞♦ ✇❡ ❝❛r❡❄

❈❛♥ ❡st✐♠❛t❡ t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❝❛❧❡❞ ♠❛①✐♠❛ s❡q✉❡♥❝❡ M(✶)

n /n ✭❛s ✐♥ ●❛❧❛♠❜♦s ✭✶✾✼✷✮✮ ✲ s✐❣♥✐✜❝❛♥t❧② ✐♠♣r♦✈❡s ❛ r❡s✉❧t

♦❢ P❤✐❧✐♣♣ ✭✶✾✼✻✮✳ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ s❝❛❧❡❞ kth ♠❛①✐♠❛ ❢♦r ❛♥② k ∈ N ✭✉♥✐❢♦r♠ ♦✈❡r k✮✳ ❆ t✐♥② ❞❡t♦✉r ♦❢ ♦✉r ♣r♦♦❢ r❡❝♦✈❡rs ❛ r❡s✉❧t ♦❢ ❚②r❛♥✲❑❛♠✐➠s❦❛ ✭✷✵✶✵✮ ♦♥ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡①tr❡♠❛❧ ♣♦✐♥t ♣r♦❝❡ss✳ ✭■♥s♣✐r❡❞ ❜② ❈❤✐❛r✐♥✐✱ ❈✐♣r✐❛♥✐ ❛♥❞ ❍❛③r❛ ✭✷✵✶✺✮✳✮ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❝❛❧❡❞ ♠❛①✐♠❛ ❢♦r t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧❛r s✉r❢❛❝❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✺ ✴ ✸✵

slide-57
SLIDE 57

❲❤② ❞♦ ✇❡ ❝❛r❡❄

❈❛♥ ❡st✐♠❛t❡ t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❝❛❧❡❞ ♠❛①✐♠❛ s❡q✉❡♥❝❡ M(✶)

n /n ✭❛s ✐♥ ●❛❧❛♠❜♦s ✭✶✾✼✷✮✮ ✲ s✐❣♥✐✜❝❛♥t❧② ✐♠♣r♦✈❡s ❛ r❡s✉❧t

♦❢ P❤✐❧✐♣♣ ✭✶✾✼✻✮✳ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ s❝❛❧❡❞ kth ♠❛①✐♠❛ ❢♦r ❛♥② k ∈ N ✭✉♥✐❢♦r♠ ♦✈❡r k✮✳ ❆ t✐♥② ❞❡t♦✉r ♦❢ ♦✉r ♣r♦♦❢ r❡❝♦✈❡rs ❛ r❡s✉❧t ♦❢ ❚②r❛♥✲❑❛♠✐➠s❦❛ ✭✷✵✶✵✮ ♦♥ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡①tr❡♠❛❧ ♣♦✐♥t ♣r♦❝❡ss✳ ✭■♥s♣✐r❡❞ ❜② ❈❤✐❛r✐♥✐✱ ❈✐♣r✐❛♥✐ ❛♥❞ ❍❛③r❛ ✭✷✵✶✺✮✳✮ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❝❛❧❡❞ ♠❛①✐♠❛ ❢♦r t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧❛r s✉r❢❛❝❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✺ ✴ ✸✵

slide-58
SLIDE 58

❲❤② ❞♦ ✇❡ ❝❛r❡❄

❈❛♥ ❡st✐♠❛t❡ t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❝❛❧❡❞ ♠❛①✐♠❛ s❡q✉❡♥❝❡ M(✶)

n /n ✭❛s ✐♥ ●❛❧❛♠❜♦s ✭✶✾✼✷✮✮ ✲ s✐❣♥✐✜❝❛♥t❧② ✐♠♣r♦✈❡s ❛ r❡s✉❧t

♦❢ P❤✐❧✐♣♣ ✭✶✾✼✻✮✳ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ s❝❛❧❡❞ kth ♠❛①✐♠❛ ❢♦r ❛♥② k ∈ N ✭✉♥✐❢♦r♠ ♦✈❡r k✮✳ ❆ t✐♥② ❞❡t♦✉r ♦❢ ♦✉r ♣r♦♦❢ r❡❝♦✈❡rs ❛ r❡s✉❧t ♦❢ ❚②r❛♥✲❑❛♠✐➠s❦❛ ✭✷✵✶✵✮ ♦♥ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡①tr❡♠❛❧ ♣♦✐♥t ♣r♦❝❡ss✳ ✭■♥s♣✐r❡❞ ❜② ❈❤✐❛r✐♥✐✱ ❈✐♣r✐❛♥✐ ❛♥❞ ❍❛③r❛ ✭✷✵✶✺✮✳✮ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❝❛❧❡❞ ♠❛①✐♠❛ ❢♦r t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧❛r s✉r❢❛❝❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✺ ✴ ✸✵

slide-59
SLIDE 59

❲❤② ❞♦ ✇❡ ❝❛r❡❄

❈❛♥ ❡st✐♠❛t❡ t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❝❛❧❡❞ ♠❛①✐♠❛ s❡q✉❡♥❝❡ M(✶)

n /n ✭❛s ✐♥ ●❛❧❛♠❜♦s ✭✶✾✼✷✮✮ ✲ s✐❣♥✐✜❝❛♥t❧② ✐♠♣r♦✈❡s ❛ r❡s✉❧t

♦❢ P❤✐❧✐♣♣ ✭✶✾✼✻✮✳ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❢♦r t❤❡ s❝❛❧❡❞ kth ♠❛①✐♠❛ ❢♦r ❛♥② k ∈ N ✭✉♥✐❢♦r♠ ♦✈❡r k✮✳ ❆ t✐♥② ❞❡t♦✉r ♦❢ ♦✉r ♣r♦♦❢ r❡❝♦✈❡rs ❛ r❡s✉❧t ♦❢ ❚②r❛♥✲❑❛♠✐➠s❦❛ ✭✷✵✶✵✮ ♦♥ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡①tr❡♠❛❧ ♣♦✐♥t ♣r♦❝❡ss✳ ✭■♥s♣✐r❡❞ ❜② ❈❤✐❛r✐♥✐✱ ❈✐♣r✐❛♥✐ ❛♥❞ ❍❛③r❛ ✭✷✵✶✺✮✳✮ ❘❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❝❛❧❡❞ ♠❛①✐♠❛ ❢♦r t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧❛r s✉r❢❛❝❡✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✺ ✴ ✸✵

slide-60
SLIDE 60

❚❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧❛r s✉r❢❛❝❡

❚❤❡ ❣r♦✉♣ SL✷(Z) := a b c d

  • : a, b, c, d ∈ Z, ad − bc = ✶
  • ❛❝ts ✐s♦♠❡tr✐❝❛❧❧② ♦♥ H := {z ∈ C : ■♠(z) > ✵} ❜② r❛t✐♦♥❛❧

tr❛♥s❢♦r♠❛t✐♦♥s✿ a b c d

  • .z = az + b

cz + d . ❙❡r✐❡s ✭✶✾✽✶✱ ✶✾✽✺✮✿ ❈♦♥♥❡❝t❡❞ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥

✷

✇✐t❤ ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠ ✉s✐♥❣ ❛ s②♠❜♦❧✐❝ ❞②♥❛♠✐❝s✳ P♦❧❧✐❝♦tt ✭✷✵✵✾✮✿ ❯s❡❞ t❤✐s ❝♦♥♥❡❝t✐♦♥ t♦ ✜♥❞ t❤❡ ✇❡❛❦ ❧✐♠✐t ♦❢ t❤❡ ♥♦r♠❛❧✐③❡❞ ♠❛①✐♠❛ ♦❢ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ ✳ ❖✉r ✇♦r❦ ②✐❡❧❞s t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ P♦❧❧✐❝♦tt✬s r❡s✉❧t✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✻ ✴ ✸✵

slide-61
SLIDE 61

❚❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧❛r s✉r❢❛❝❡

❚❤❡ ❣r♦✉♣ SL✷(Z) := a b c d

  • : a, b, c, d ∈ Z, ad − bc = ✶
  • ❛❝ts ✐s♦♠❡tr✐❝❛❧❧② ♦♥ H := {z ∈ C : ■♠(z) > ✵} ❜② r❛t✐♦♥❛❧

tr❛♥s❢♦r♠❛t✐♦♥s✿ a b c d

  • .z = az + b

cz + d . ❙❡r✐❡s ✭✶✾✽✶✱ ✶✾✽✺✮✿ ❈♦♥♥❡❝t❡❞ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ M = H/SL✷(Z) ✇✐t❤ ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠ ✉s✐♥❣ ❛ s②♠❜♦❧✐❝ ❞②♥❛♠✐❝s✳ P♦❧❧✐❝♦tt ✭✷✵✵✾✮✿ ❯s❡❞ t❤✐s ❝♦♥♥❡❝t✐♦♥ t♦ ✜♥❞ t❤❡ ✇❡❛❦ ❧✐♠✐t ♦❢ t❤❡ ♥♦r♠❛❧✐③❡❞ ♠❛①✐♠❛ ♦❢ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ ✳ ❖✉r ✇♦r❦ ②✐❡❧❞s t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ P♦❧❧✐❝♦tt✬s r❡s✉❧t✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✻ ✴ ✸✵

slide-62
SLIDE 62

❚❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧❛r s✉r❢❛❝❡

❚❤❡ ❣r♦✉♣ SL✷(Z) := a b c d

  • : a, b, c, d ∈ Z, ad − bc = ✶
  • ❛❝ts ✐s♦♠❡tr✐❝❛❧❧② ♦♥ H := {z ∈ C : ■♠(z) > ✵} ❜② r❛t✐♦♥❛❧

tr❛♥s❢♦r♠❛t✐♦♥s✿ a b c d

  • .z = az + b

cz + d . ❙❡r✐❡s ✭✶✾✽✶✱ ✶✾✽✺✮✿ ❈♦♥♥❡❝t❡❞ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ M = H/SL✷(Z) ✇✐t❤ ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠ ✉s✐♥❣ ❛ s②♠❜♦❧✐❝ ❞②♥❛♠✐❝s✳ P♦❧❧✐❝♦tt ✭✷✵✵✾✮✿ ❯s❡❞ t❤✐s ❝♦♥♥❡❝t✐♦♥ t♦ ✜♥❞ t❤❡ ✇❡❛❦ ❧✐♠✐t ♦❢ t❤❡ ♥♦r♠❛❧✐③❡❞ ♠❛①✐♠❛ ♦❢ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ M✳ ❖✉r ✇♦r❦ ②✐❡❧❞s t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ P♦❧❧✐❝♦tt✬s r❡s✉❧t✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✻ ✴ ✸✵

slide-63
SLIDE 63

❚❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ t❤❡ ♠♦❞✉❧❛r s✉r❢❛❝❡

❚❤❡ ❣r♦✉♣ SL✷(Z) := a b c d

  • : a, b, c, d ∈ Z, ad − bc = ✶
  • ❛❝ts ✐s♦♠❡tr✐❝❛❧❧② ♦♥ H := {z ∈ C : ■♠(z) > ✵} ❜② r❛t✐♦♥❛❧

tr❛♥s❢♦r♠❛t✐♦♥s✿ a b c d

  • .z = az + b

cz + d . ❙❡r✐❡s ✭✶✾✽✶✱ ✶✾✽✺✮✿ ❈♦♥♥❡❝t❡❞ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ M = H/SL✷(Z) ✇✐t❤ ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠ ✉s✐♥❣ ❛ s②♠❜♦❧✐❝ ❞②♥❛♠✐❝s✳ P♦❧❧✐❝♦tt ✭✷✵✵✾✮✿ ❯s❡❞ t❤✐s ❝♦♥♥❡❝t✐♦♥ t♦ ✜♥❞ t❤❡ ✇❡❛❦ ❧✐♠✐t ♦❢ t❤❡ ♥♦r♠❛❧✐③❡❞ ♠❛①✐♠❛ ♦❢ t❤❡ ❣❡♦❞❡s✐❝ ✢♦✇ ♦♥ M✳ ❖✉r ✇♦r❦ ②✐❡❧❞s t❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ P♦❧❧✐❝♦tt✬s r❡s✉❧t✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✻ ✴ ✸✵

slide-64
SLIDE 64

❚❤❡ ♠❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠ ✭●❤♦s❤✱ ❑✐rs❡❜♦♠✱ ❘✳ ✭✷✵✶✾✮✮

❚❤❡r❡ ❡①✐sts κ > ✵ ❛♥❞ ❛ s❡q✉❡♥❝❡ ✶ ≪ ℓn ≪ nǫ ✭❢♦r ❛❧❧ ǫ > ✵✮ s✉❝❤ t❤❛t ❢♦r ❛❧❧ u > ✵ ❛♥❞ ❢♦r ❛❧❧ n ∈ N✱ dTV (Eu

n , Eu ∞) :=

sup

A⊆N∪{✵}

  • P(Eu

n ∈ A)−P(Eu ∞ ∈ A)

  • ≤

κ min {u, u✷} ℓn n .

❈♦r♦❧❧❛r②

❙✉♣♣♦s❡ ♠❛①✐♠✉♠ ♦❢ ✷ ✶ ✳ ❋♦r ❛❧❧ ✵ ❛♥❞ ❢♦r ❛❧❧ ✱

✶

✶ ✵ ✷

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✼ ✴ ✸✵

slide-65
SLIDE 65

❚❤❡ ♠❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠ ✭●❤♦s❤✱ ❑✐rs❡❜♦♠✱ ❘✳ ✭✷✵✶✾✮✮

❚❤❡r❡ ❡①✐sts κ > ✵ ❛♥❞ ❛ s❡q✉❡♥❝❡ ✶ ≪ ℓn ≪ nǫ ✭❢♦r ❛❧❧ ǫ > ✵✮ s✉❝❤ t❤❛t ❢♦r ❛❧❧ u > ✵ ❛♥❞ ❢♦r ❛❧❧ n ∈ N✱ dTV (Eu

n , Eu ∞) :=

sup

A⊆N∪{✵}

  • P(Eu

n ∈ A)−P(Eu ∞ ∈ A)

  • ≤

κ min {u, u✷} ℓn n .

❈♦r♦❧❧❛r②

❙✉♣♣♦s❡ M(k)

n

:= kth ♠❛①✐♠✉♠ ♦❢ {Ai log ✷ : ✶ ≤ i ≤ n}✳ ❋♦r ❛❧❧ u > ✵ ❛♥❞ ❢♦r ❛❧❧ k, n ∈ N✱ sup

k∈N

  • P
  • M(k)

n

n ≤ u

  • − e−u−✶

k−✶

  • i=✵

u−i i!

  • ≤

κ min {u, u✷} ℓn n .

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✼ ✴ ✸✵

slide-66
SLIDE 66

❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❡①✐st✐♥❣ r❡s✉❧ts

❘❡s♥✐❝❦ ❛♥❞ ❞❡ ❍❛❛♥ ✭✶✾✽✾✮✿ ■❢ A✶, A✷, . . . ✇❡r❡ ✐♥❞❡♣❡♥❞❡♥t✱ t❤❡♥

  • P
  • M(✶)

n /n ≤ u

  • − e−u−✶
  • ≤ O(✶/n).

❖✉r ✉♣♣❡r ❜♦✉♥❞ ❂

✶

❢♦r ❛❧❧ ✵✳ P❤✐❧✐♣♣ ✭✶✾✼✻✮✿ ❋♦r ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠

✶

✶

❢♦r ❛❧❧ ✵ ✶ ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✽ ✴ ✸✵

slide-67
SLIDE 67

❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❡①✐st✐♥❣ r❡s✉❧ts

❘❡s♥✐❝❦ ❛♥❞ ❞❡ ❍❛❛♥ ✭✶✾✽✾✮✿ ■❢ A✶, A✷, . . . ✇❡r❡ ✐♥❞❡♣❡♥❞❡♥t✱ t❤❡♥

  • P
  • M(✶)

n /n ≤ u

  • − e−u−✶
  • ≤ O(✶/n).

❖✉r ✉♣♣❡r ❜♦✉♥❞ ❂ O(ℓn/n) = o

  • n−✶+ǫ

❢♦r ❛❧❧ ǫ > ✵✳ P❤✐❧✐♣♣ ✭✶✾✼✻✮✿ ❋♦r ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠

✶

✶

❢♦r ❛❧❧ ✵ ✶ ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✽ ✴ ✸✵

slide-68
SLIDE 68

❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❡①✐st✐♥❣ r❡s✉❧ts

❘❡s♥✐❝❦ ❛♥❞ ❞❡ ❍❛❛♥ ✭✶✾✽✾✮✿ ■❢ A✶, A✷, . . . ✇❡r❡ ✐♥❞❡♣❡♥❞❡♥t✱ t❤❡♥

  • P
  • M(✶)

n /n ≤ u

  • − e−u−✶
  • ≤ O(✶/n).

❖✉r ✉♣♣❡r ❜♦✉♥❞ ❂ O(ℓn/n) = o

  • n−✶+ǫ

❢♦r ❛❧❧ ǫ > ✵✳ P❤✐❧✐♣♣ ✭✶✾✼✻✮✿ ❋♦r ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠

  • P
  • M(✶)

n /n ≤ u

  • − e−u−✶
  • ≤ O(ℓn/n) ≪O
  • exp
  • −(log n)δ

❢♦r ❛❧❧ δ ∈ (✵, ✶)✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✽ ✴ ✸✵

slide-69
SLIDE 69

❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❡①✐st✐♥❣ r❡s✉❧ts

❘❡s♥✐❝❦ ❛♥❞ ❞❡ ❍❛❛♥ ✭✶✾✽✾✮✿ ■❢ A✶, A✷, . . . ✇❡r❡ ✐♥❞❡♣❡♥❞❡♥t✱ t❤❡♥

  • P
  • M(✶)

n /n ≤ u

  • − e−u−✶
  • ≤ O(✶/n).

❖✉r ✉♣♣❡r ❜♦✉♥❞ ❂ O(ℓn/n) = o

  • n−✶+ǫ

❢♦r ❛❧❧ ǫ > ✵✳ P❤✐❧✐♣♣ ✭✶✾✼✻✮✿ ❋♦r ●❛✉ss ❞②♥❛♠✐❝❛❧ s②st❡♠

  • P
  • M(✶)

n /n ≤ u

  • − e−u−✶
  • ≤ O(ℓn/n) ≪O
  • exp
  • −(log n)δ

❢♦r ❛❧❧ δ ∈ (✵, ✶)✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✶✾ ✴ ✸✵

slide-70
SLIDE 70

❙❦❡t❝❤ ♦❢ ♣r♦♦❢

❘❡❝❛❧❧ Eu

n = n j=✶ ✶(Aj log ✷>un) approx

∼ Bin

  • n, pn = P(A✶ log ✷ > un)
  • ✳

❉❡✜♥❡ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ˜ Eu

n ∼ Poi(npn)✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ Eu

∞ ∼ Poi(u−✶)✳

  • ❯s❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②

dTV (Eu

n , Eu ∞) ≤ dTV (Eu n , ˜

Eu

n ) + dTV ( ˜

Eu

n , Eu ∞).

  • ❇♦✉♥❞ dTV (Eu

n , ˜

Eu

n ) ✉s✐♥❣ ❈❤❡♥✲❙t❡✐♥ ♠❡t❤♦❞ ✭❆rr❛t✐❛✱ ●♦❧❞st❡✐♥

❛♥❞ ●♦r❞♦♥ ✭✶✾✽✾✮✮ ✰ ❡①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣ ✭P❤✐❧✐♣♣ ✭✶✾✼✵✮✮✳

  • ❊st✐♠❛t❡ dTV ( ˜

Eu

n , Eu ∞) ✉s✐♥❣ s❡❝♦♥❞ ♦r❞❡r r❡❣✉❧❛r ✈❛r✐❛t✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✵ ✴ ✸✵

slide-71
SLIDE 71

❙❦❡t❝❤ ♦❢ ♣r♦♦❢

❘❡❝❛❧❧ Eu

n = n j=✶ ✶(Aj log ✷>un) approx

∼ Bin

  • n, pn = P(A✶ log ✷ > un)
  • ✳

❉❡✜♥❡ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ˜ Eu

n ∼ Poi(npn)✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ Eu

∞ ∼ Poi(u−✶)✳

  • ❯s❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②

dTV (Eu

n , Eu ∞) ≤ dTV (Eu n , ˜

Eu

n ) + dTV ( ˜

Eu

n , Eu ∞).

  • ❇♦✉♥❞ dTV (Eu

n , ˜

Eu

n ) ✉s✐♥❣ ❈❤❡♥✲❙t❡✐♥ ♠❡t❤♦❞ ✭❆rr❛t✐❛✱ ●♦❧❞st❡✐♥

❛♥❞ ●♦r❞♦♥ ✭✶✾✽✾✮✮ ✰ ❡①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣ ✭P❤✐❧✐♣♣ ✭✶✾✼✵✮✮✳

  • ❊st✐♠❛t❡ dTV ( ˜

Eu

n , Eu ∞) ✉s✐♥❣ s❡❝♦♥❞ ♦r❞❡r r❡❣✉❧❛r ✈❛r✐❛t✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✶ ✴ ✸✵

slide-72
SLIDE 72

❙❦❡t❝❤ ♦❢ ♣r♦♦❢

❘❡❝❛❧❧ Eu

n = n j=✶ ✶(Aj log ✷>un) approx

∼ Bin

  • n, pn = P(A✶ log ✷ > un)
  • ✳

❉❡✜♥❡ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ˜ Eu

n ∼ Poi(npn)✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ Eu

∞ ∼ Poi(u−✶)✳

  • ❯s❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②

dTV (Eu

n , Eu ∞) ≤ dTV (Eu n , ˜

Eu

n ) + dTV ( ˜

Eu

n , Eu ∞).

  • ❇♦✉♥❞ dTV (Eu

n , ˜

Eu

n ) ✉s✐♥❣ ❈❤❡♥✲❙t❡✐♥ ♠❡t❤♦❞ ✭❆rr❛t✐❛✱ ●♦❧❞st❡✐♥

❛♥❞ ●♦r❞♦♥ ✭✶✾✽✾✮✮ ✰ ❡①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣ ✭P❤✐❧✐♣♣ ✭✶✾✼✵✮✮✳

  • ❊st✐♠❛t❡ dTV ( ˜

Eu

n , Eu ∞) ✉s✐♥❣ s❡❝♦♥❞ ♦r❞❡r r❡❣✉❧❛r ✈❛r✐❛t✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✷ ✴ ✸✵

slide-73
SLIDE 73

❙❦❡t❝❤ ♦❢ ♣r♦♦❢

❘❡❝❛❧❧ Eu

n = n j=✶ ✶(Aj log ✷>un) approx

∼ Bin

  • n, pn = P(A✶ log ✷ > un)
  • ✳

❉❡✜♥❡ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ˜ Eu

n ∼ Poi(npn)✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ Eu

∞ ∼ Poi(u−✶)✳

  • ❯s❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②

dTV (Eu

n , Eu ∞) ≤ dTV (Eu n , ˜

Eu

n ) + dTV ( ˜

Eu

n , Eu ∞).

  • ❇♦✉♥❞ dTV (Eu

n , ˜

Eu

n ) ✉s✐♥❣ ❈❤❡♥✲❙t❡✐♥ ♠❡t❤♦❞ ✭❆rr❛t✐❛✱ ●♦❧❞st❡✐♥

❛♥❞ ●♦r❞♦♥ ✭✶✾✽✾✮✮ ✰ ❡①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣ ✭P❤✐❧✐♣♣ ✭✶✾✼✵✮✮✳

  • ❊st✐♠❛t❡ dTV ( ˜

Eu

n , Eu ∞) ✉s✐♥❣ s❡❝♦♥❞ ♦r❞❡r r❡❣✉❧❛r ✈❛r✐❛t✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✸ ✴ ✸✵

slide-74
SLIDE 74

❙❦❡t❝❤ ♦❢ ♣r♦♦❢

❘❡❝❛❧❧ Eu

n = n j=✶ ✶(Aj log ✷>un) approx

∼ Bin

  • n, pn = P(A✶ log ✷ > un)
  • ✳

❉❡✜♥❡ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ˜ Eu

n ∼ Poi(npn)✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ Eu

∞ ∼ Poi(u−✶)✳

  • ❯s❡ tr✐❛♥❣❧❡ ✐♥❡q✉❛❧✐t②

dTV (Eu

n , Eu ∞) ≤ dTV (Eu n , ˜

Eu

n ) + dTV ( ˜

Eu

n , Eu ∞).

  • ❇♦✉♥❞ dTV (Eu

n , ˜

Eu

n ) ✉s✐♥❣ ❈❤❡♥✲❙t❡✐♥ ♠❡t❤♦❞ ✭❆rr❛t✐❛✱ ●♦❧❞st❡✐♥

❛♥❞ ●♦r❞♦♥ ✭✶✾✽✾✮✮ ✰ ❡①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣ ✭P❤✐❧✐♣♣ ✭✶✾✼✵✮✮✳

  • ❊st✐♠❛t❡ dTV ( ˜

Eu

n , Eu ∞) ✉s✐♥❣ s❡❝♦♥❞ ♦r❞❡r r❡❣✉❧❛r ✈❛r✐❛t✐♦♥✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✹ ✴ ✸✵

slide-75
SLIDE 75

❍♦✇ t♦ ❡st✐♠❛t❡ dTV( ˜ Eu

n, Eu ∞)❄ ❘❡❝❛❧❧ ˜ Eu

n ∼ Poi(npn) ❛♥❞

Eu

∞ ∼ Poi(u−✶)✳

▲❡♠♠❛ ✭✽✮ ♦❢ ❋r❡❡❞♠❛♥ ✭✶✾✼✹✮✿

✶

✭s♦❢t ❜♦✉♥❞✮

✶

✷

✶

✸ ✷ ✷

✷

✶ ✭s❡❝♦♥❞ ♦r❞❡r r❡❣✉❧❛r ✈❛r✐❛t✐♦♥✮

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✺ ✴ ✸✵

slide-76
SLIDE 76

❍♦✇ t♦ ❡st✐♠❛t❡ dTV( ˜ Eu

n, Eu ∞)❄ ❘❡❝❛❧❧ ˜ Eu

n ∼ Poi(npn) ❛♥❞

Eu

∞ ∼ Poi(u−✶)✳

▲❡♠♠❛ ✭✽✮ ♦❢ ❋r❡❡❞♠❛♥ ✭✶✾✼✹✮✿ dTV ( ˜ Eu

n , Eu ∞) ≤

  • npn − u−✶
  • ✭s♦❢t ❜♦✉♥❞✮

=

  • nP(A✶ log ✷ > un) − u−✶
  • ✸

✷ ✷

✷

✶ ✭s❡❝♦♥❞ ♦r❞❡r r❡❣✉❧❛r ✈❛r✐❛t✐♦♥✮

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✺ ✴ ✸✵

slide-77
SLIDE 77

❍♦✇ t♦ ❡st✐♠❛t❡ dTV( ˜ Eu

n, Eu ∞)❄ ❘❡❝❛❧❧ ˜ Eu

n ∼ Poi(npn) ❛♥❞

Eu

∞ ∼ Poi(u−✶)✳

▲❡♠♠❛ ✭✽✮ ♦❢ ❋r❡❡❞♠❛♥ ✭✶✾✼✹✮✿ dTV ( ˜ Eu

n , Eu ∞) ≤

  • npn − u−✶
  • ✭s♦❢t ❜♦✉♥❞✮

=

  • nP(A✶ log ✷ > un) − u−✶
  • ≤ ✸ log ✷

✷u✷ ✶ n ✭s❡❝♦♥❞ ♦r❞❡r r❡❣✉❧❛r ✈❛r✐❛t✐♦♥✮

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✺ ✴ ✸✵

slide-78
SLIDE 78

❍♦✇ t♦ ❡st✐♠❛t❡ dTV( ˜ Eu

n, Eu ∞)❄ ❘❡❝❛❧❧ ˜ Eu

n ∼ Poi(npn) ❛♥❞

Eu

∞ ∼ Poi(u−✶)✳

▲❡♠♠❛ ✭✽✮ ♦❢ ❋r❡❡❞♠❛♥ ✭✶✾✼✹✮✿ dTV ( ˜ Eu

n , Eu ∞) ≤

  • npn − u−✶
  • ✭s♦❢t ❜♦✉♥❞✮

=

  • nP(A✶ log ✷ > un) − u−✶
  • ≤ ✸ log ✷

✷u✷ ✶ n ✭s❡❝♦♥❞ ♦r❞❡r r❡❣✉❧❛r ✈❛r✐❛t✐♦♥✮ ≪ ℓn n .

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✺ ✴ ✸✵

slide-79
SLIDE 79

❍♦✇ t♦ ❜♦✉♥❞ dTV(Eu

n, ˜

Eu

n)❄ ❘❡❝❛❧❧ Eu

n = n i=✶ ✶(Ai log ✷>un)

❛♥❞ ˜ Eu

n ∼ Poi(npn)✳

✶ ✷ ✳ ✶

✷

✭❞❡♣❡♥❞❡♥t✮✳ ◆♦t❡ t❤❛t ✳ ❚❛❦❡

✶ ✷

✳ ❚❤❡r❡❢♦r❡ ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✻ ✴ ✸✵

slide-80
SLIDE 80

❍♦✇ t♦ ❜♦✉♥❞ dTV(Eu

n, ˜

Eu

n)❄ ❘❡❝❛❧❧ Eu

n = n i=✶ ✶(Ai log ✷>un)

❛♥❞ ˜ Eu

n ∼ Poi(npn)✳

I := {✶, ✷, . . . , n}✳ ✶

✷

✭❞❡♣❡♥❞❡♥t✮✳ ◆♦t❡ t❤❛t ✳ ❚❛❦❡

✶ ✷

✳ ❚❤❡r❡❢♦r❡ ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✻ ✴ ✸✵

slide-81
SLIDE 81

❍♦✇ t♦ ❜♦✉♥❞ dTV(Eu

n, ˜

Eu

n)❄ ❘❡❝❛❧❧ Eu

n = n i=✶ ✶(Ai log ✷>un)

❛♥❞ ˜ Eu

n ∼ Poi(npn)✳

I := {✶, ✷, . . . , n}✳ {Xi := ✶(Ai log ✷>un) ∼ Ber(pn)}i∈I ✭❞❡♣❡♥❞❡♥t✮✳ ◆♦t❡ t❤❛t ✳ ❚❛❦❡

✶ ✷

✳ ❚❤❡r❡❢♦r❡ ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✻ ✴ ✸✵

slide-82
SLIDE 82

❍♦✇ t♦ ❜♦✉♥❞ dTV(Eu

n, ˜

Eu

n)❄ ❘❡❝❛❧❧ Eu

n = n i=✶ ✶(Ai log ✷>un)

❛♥❞ ˜ Eu

n ∼ Poi(npn)✳

I := {✶, ✷, . . . , n}✳ {Xi := ✶(Ai log ✷>un) ∼ Ber(pn)}i∈I ✭❞❡♣❡♥❞❡♥t✮✳ ◆♦t❡ t❤❛t Eu

n = i∈I Xi✳

❚❛❦❡

✶ ✷

✳ ❚❤❡r❡❢♦r❡ ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✻ ✴ ✸✵

slide-83
SLIDE 83

❍♦✇ t♦ ❜♦✉♥❞ dTV(Eu

n, ˜

Eu

n)❄ ❘❡❝❛❧❧ Eu

n = n i=✶ ✶(Ai log ✷>un)

❛♥❞ ˜ Eu

n ∼ Poi(npn)✳

I := {✶, ✷, . . . , n}✳ {Xi := ✶(Ai log ✷>un) ∼ Ber(pn)}i∈I ✭❞❡♣❡♥❞❡♥t✮✳ ◆♦t❡ t❤❛t Eu

n = i∈I Xi✳

❚❛❦❡ Y✶, Y✷, . . . , Yn

iid

∼ Poi(pn)✳ ❚❤❡r❡❢♦r❡ ✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✻ ✴ ✸✵

slide-84
SLIDE 84

❍♦✇ t♦ ❜♦✉♥❞ dTV(Eu

n, ˜

Eu

n)❄ ❘❡❝❛❧❧ Eu

n = n i=✶ ✶(Ai log ✷>un)

❛♥❞ ˜ Eu

n ∼ Poi(npn)✳

I := {✶, ✷, . . . , n}✳ {Xi := ✶(Ai log ✷>un) ∼ Ber(pn)}i∈I ✭❞❡♣❡♥❞❡♥t✮✳ ◆♦t❡ t❤❛t Eu

n = i∈I Xi✳

❚❛❦❡ Y✶, Y✷, . . . , Yn

iid

∼ Poi(pn)✳ ❚❤❡r❡❢♦r❡ ˜ Eu

n L

=

i∈I Yi✳

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✻ ✴ ✸✵

slide-85
SLIDE 85

❍♦✇ t♦ ❜♦✉♥❞ dTV(Eu

n, ˜

Eu

n)❄ ❘❡❝❛❧❧ Eu

n = n i=✶ ✶(Ai log ✷>un)

❛♥❞ ˜ Eu

n ∼ Poi(npn)✳

I := {✶, ✷, . . . , n}✳ {Xi := ✶(Ai log ✷>un) ∼ Ber(pn)}i∈I ✭❞❡♣❡♥❞❡♥t✮✳ ◆♦t❡ t❤❛t Eu

n = i∈I Xi✳

❚❛❦❡ Y✶, Y✷, . . . , Yn

iid

∼ Poi(pn)✳ ❚❤❡r❡❢♦r❡ ˜ Eu

n L

=

i∈I Yi✳

dTV (Eu

n , ˜

Eu

n ) = dTV i∈I

Xi ,

  • i∈I

Yi

  • ≤ ??

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✻ ✴ ✸✵

slide-86
SLIDE 86

❆rr❛t✐❛✱ ●♦❧❞st❡✐♥ ❛♥❞ ●♦r❞♦♥ ✭✶✾✽✾✮

{Xi ∼ Ber(πi)}i∈I ✭♣♦ss✐❜❧② ❞❡♣❡♥❞❡♥t✮✳ {Yi ∼ Poi(πi)}i∈I ✭✐♥❞❡♣❡♥❞❡♥t✮✳ ❋♦r ❡❛❝❤ ✱ t❤❡r❡ ❡①✐sts ❛ s✉❜s❡t s✉❝❤ t❤❛t ❛♥❞ ✐s ✏♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t✑ ♦❢ ✳

✶ ✷ ✸

✹ ✶ ✹ ✷ ✷ ✸

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✼ ✴ ✸✵

slide-87
SLIDE 87

❆rr❛t✐❛✱ ●♦❧❞st❡✐♥ ❛♥❞ ●♦r❞♦♥ ✭✶✾✽✾✮

{Xi ∼ Ber(πi)}i∈I ✭♣♦ss✐❜❧② ❞❡♣❡♥❞❡♥t✮✳ {Yi ∼ Poi(πi)}i∈I ✭✐♥❞❡♣❡♥❞❡♥t✮✳ ❋♦r ❡❛❝❤ i ∈ I✱ t❤❡r❡ ❡①✐sts ❛ s✉❜s❡t Bi ⊆ I s✉❝❤ t❤❛t i ∈ Bi ❛♥❞ Xi ✐s ✏♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t✑ ♦❢ {Xj : j ∈ I \ Bi}✳

✶ ✷ ✸

✹ ✶ ✹ ✷ ✷ ✸

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✼ ✴ ✸✵

slide-88
SLIDE 88

❆rr❛t✐❛✱ ●♦❧❞st❡✐♥ ❛♥❞ ●♦r❞♦♥ ✭✶✾✽✾✮

{Xi ∼ Ber(πi)}i∈I ✭♣♦ss✐❜❧② ❞❡♣❡♥❞❡♥t✮✳ {Yi ∼ Poi(πi)}i∈I ✭✐♥❞❡♣❡♥❞❡♥t✮✳ ❋♦r ❡❛❝❤ i ∈ I✱ t❤❡r❡ ❡①✐sts ❛ s✉❜s❡t Bi ⊆ I s✉❝❤ t❤❛t i ∈ Bi ❛♥❞ Xi ✐s ✏♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t✑ ♦❢ {Xj : j ∈ I \ Bi}✳ b✶ :=

  • i∈I
  • j∈Bi

πiπj, b✷ :=

  • i∈I
  • j∈Bi\{i}

E(XiXj),

✸

✹ ✶ ✹ ✷ ✷ ✸

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✼ ✴ ✸✵

slide-89
SLIDE 89

❆rr❛t✐❛✱ ●♦❧❞st❡✐♥ ❛♥❞ ●♦r❞♦♥ ✭✶✾✽✾✮

{Xi ∼ Ber(πi)}i∈I ✭♣♦ss✐❜❧② ❞❡♣❡♥❞❡♥t✮✳ {Yi ∼ Poi(πi)}i∈I ✭✐♥❞❡♣❡♥❞❡♥t✮✳ ❋♦r ❡❛❝❤ i ∈ I✱ t❤❡r❡ ❡①✐sts ❛ s✉❜s❡t Bi ⊆ I s✉❝❤ t❤❛t i ∈ Bi ❛♥❞ Xi ✐s ✏♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t✑ ♦❢ {Xj : j ∈ I \ Bi}✳ b✶ :=

  • i∈I
  • j∈Bi

πiπj, b✷ :=

  • i∈I
  • j∈Bi\{i}

E(XiXj), b✸ :=

  • i∈I

E

  • E(Xi − πi
  • {Xj : j ∈ I \ Bj}
  • .

✹ ✶ ✹ ✷ ✷ ✸

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✼ ✴ ✸✵

slide-90
SLIDE 90

❆rr❛t✐❛✱ ●♦❧❞st❡✐♥ ❛♥❞ ●♦r❞♦♥ ✭✶✾✽✾✮

{Xi ∼ Ber(πi)}i∈I ✭♣♦ss✐❜❧② ❞❡♣❡♥❞❡♥t✮✳ {Yi ∼ Poi(πi)}i∈I ✭✐♥❞❡♣❡♥❞❡♥t✮✳ ❋♦r ❡❛❝❤ i ∈ I✱ t❤❡r❡ ❡①✐sts ❛ s✉❜s❡t Bi ⊆ I s✉❝❤ t❤❛t i ∈ Bi ❛♥❞ Xi ✐s ✏♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t✑ ♦❢ {Xj : j ∈ I \ Bi}✳ b✶ :=

  • i∈I
  • j∈Bi

πiπj, b✷ :=

  • i∈I
  • j∈Bi\{i}

E(XiXj), b✸ :=

  • i∈I

E

  • E(Xi − πi
  • {Xj : j ∈ I \ Bj}
  • .

dTV

i∈I

Xi ,

  • i∈I

Yi

  • ≤ ✹b✶ + ✹b✷ + ✷b✸

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✼ ✴ ✸✵

slide-91
SLIDE 91

❆rr❛t✐❛✱ ●♦❧❞st❡✐♥ ❛♥❞ ●♦r❞♦♥ ✭✶✾✽✾✮

{Xi ∼ Ber(πi)}i∈I ✭♣♦ss✐❜❧② ❞❡♣❡♥❞❡♥t✮✳ {Yi ∼ Poi(πi)}i∈I ✭✐♥❞❡♣❡♥❞❡♥t✮✳ ❋♦r ❡❛❝❤ i ∈ I✱ t❤❡r❡ ❡①✐sts ❛ s✉❜s❡t Bi ⊆ I s✉❝❤ t❤❛t i ∈ Bi ❛♥❞ Xi ✐s ✏♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t✑ ♦❢ {Xj : j ∈ I \ Bi}✳ b✶ :=

  • i∈I
  • j∈Bi

πiπj, b✷ :=

  • i∈I
  • j∈Bi\{i}

E(XiXj), b✸ :=

  • i∈I

E

  • E(Xi − πi
  • {Xj : j ∈ I \ Bj}
  • .

dTV (Eu

n , ˜

Eu

n ) = dTV i∈I

Xi ,

  • i∈I

Yi

  • ≤ ✹b✶ + ✹b✷ + ✷b✸

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✽ ✴ ✸✵

slide-92
SLIDE 92

❍♦✇ t♦ ❜♦✉♥❞ b✶✱ b✷✱ b✸❄

❆❢t❡r ❝❛r❡❢✉❧❧② ❝❤♦♦s✐♥❣ t❤❡ Bi✬s ✳ ✳ ✳

✶ ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ ❡❛s✐❧②✱ ❛♥❞

t❤❡ ❜♦✉♥❞s ♦♥

✷ ❛♥❞ ✸ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣

♣r♦♣❡rt② ♦❢ ✬s✿

❚❤❡♦r❡♠ ✭P❤✐❧✐♣♣ ✭✶✾✼✵✮✮

❚❤❡r❡ ❡①✐sts ✵ ❛♥❞ ✶ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✱ ❢♦r ❛❧❧

✶ ✷

✱ ❛♥❞ ❢♦r ❛❧❧

✶

✱

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✾ ✴ ✸✵

slide-93
SLIDE 93

❍♦✇ t♦ ❜♦✉♥❞ b✶✱ b✷✱ b✸❄

❆❢t❡r ❝❛r❡❢✉❧❧② ❝❤♦♦s✐♥❣ t❤❡ Bi✬s ✳ ✳ ✳ b✶ ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ ❡❛s✐❧②✱ ❛♥❞ t❤❡ ❜♦✉♥❞s ♦♥

✷ ❛♥❞ ✸ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣

♣r♦♣❡rt② ♦❢ ✬s✿

❚❤❡♦r❡♠ ✭P❤✐❧✐♣♣ ✭✶✾✼✵✮✮

❚❤❡r❡ ❡①✐sts ✵ ❛♥❞ ✶ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✱ ❢♦r ❛❧❧

✶ ✷

✱ ❛♥❞ ❢♦r ❛❧❧

✶

✱

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✾ ✴ ✸✵

slide-94
SLIDE 94

❍♦✇ t♦ ❜♦✉♥❞ b✶✱ b✷✱ b✸❄

❆❢t❡r ❝❛r❡❢✉❧❧② ❝❤♦♦s✐♥❣ t❤❡ Bi✬s ✳ ✳ ✳ b✶ ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ ❡❛s✐❧②✱ ❛♥❞ t❤❡ ❜♦✉♥❞s ♦♥ b✷ ❛♥❞ b✸ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣ ♣r♦♣❡rt② ♦❢ Ai✬s✿

❚❤❡♦r❡♠ ✭P❤✐❧✐♣♣ ✭✶✾✼✵✮✮

❚❤❡r❡ ❡①✐sts ✵ ❛♥❞ ✶ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ✱ ❢♦r ❛❧❧

✶ ✷

✱ ❛♥❞ ❢♦r ❛❧❧

✶

✱

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✾ ✴ ✸✵

slide-95
SLIDE 95

❍♦✇ t♦ ❜♦✉♥❞ b✶✱ b✷✱ b✸❄

❆❢t❡r ❝❛r❡❢✉❧❧② ❝❤♦♦s✐♥❣ t❤❡ Bi✬s ✳ ✳ ✳ b✶ ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ ❡❛s✐❧②✱ ❛♥❞ t❤❡ ❜♦✉♥❞s ♦♥ b✷ ❛♥❞ b✸ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣♦♥❡♥t✐❛❧ ♠✐①✐♥❣ ♣r♦♣❡rt② ♦❢ Ai✬s✿

❚❤❡♦r❡♠ ✭P❤✐❧✐♣♣ ✭✶✾✼✵✮✮

❚❤❡r❡ ❡①✐sts C > ✵ ❛♥❞ θ > ✶ s✉❝❤ t❤❛t ❢♦r ❛❧❧ m, n ∈ N✱ ❢♦r ❛❧❧ F ∈ σ(A✶, A✷, . . . , Am)✱ ❛♥❞ ❢♦r ❛❧❧ H ∈ σ(Am+n, Am+n+✶, . . .)✱ |P(F ∩ H) − P(F)P(H)| ≤ Cθ−n P(F)P(H).

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✷✾ ✴ ✸✵

slide-96
SLIDE 96

❚❤❛♥❦ ❨♦✉ ❱❡r② ▼✉❝❤

P❛rt❤❛♥✐❧ ❘♦② ❈❋ ❛♥❞ ❊❱❚ ❆♣r✐❧ ✷✺✱ ✷✵✶✾ ✸✵ ✴ ✸✵