strt tr rtrs t - - PowerPoint PPT Presentation

❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ✈✐❛ t❤❡ ♠❛rt✐♥❣❛❧❡ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮

❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆✳ ❙♦✉t❤✇❡❧❧ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❛♥❞ ▼✳ ❩❤✉❦♦✈s❦✐✐ ✭▼♦s❝♦✇ ■♥st✐t✉t❡ ♦❢ P❤②s✐❝s ❛♥❞ ❚❡❝❤♥♦❧♦❣②✮ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s ●r♦✉♣ t❛❧❦✱ ❆♣r✐❧ ✷✾✱ ✷✵✶✾

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❈▲❚ ❢♦r tr❡❡ ♣❛r❛♠❡t❡rs

❉❡♥♦t❡ ❜② Tn t❤❡ s❡t ♦❢ ❛❧❧ ❧❛❜❡❧❧❡❞ tr❡❡s ✇✐t❤ ✈❡rt✐❝❡s {1, . . . , n}✳ ▲❡t T ❜❡ ❛ ✉♥✐❢♦r♠ r❛♥❞♦♠ ❡❧❡♠❡♥t ♦❢ Tn✳ ■❢ F : Tn → R ✐s ✏♥✐❝❡✑ t❤❡♥ F (T) ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧✱ ✐✳❡✳ Pr F (T) − µ σ ≤ t

• n → ∞

− → Φ(t) =

1 √ 2π

t

−∞

e−x2/2dx, ✇❤❡r❡ µ = EF (T) ❛♥❞ σ2 = Var F (T).

❲❡ ❛♥❛❧②③❡ ❛❧s♦ t❤❡ r❛t❡s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ δK(F ) =

• CDF

F (T)−µ

σ

• − Φ
• ∞ t♦ ③❡r♦✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✷ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❚r❡❡ ♣❡rt✉r❜❛t✐♦♥s

❉❡✜♥❡ ❛ tr❡❡ ♣❡rt✉r❜❛t✐♦♥ ❜② Sjk

i

T = T − ij + ik. P✐❝t✉r❡ ✇✐❧❧ ❜❡ ❤❡r❡ ❲❡ s❛② ✐s ✲▲✐♣s❝❤✐t③ ♦♥ ✐❢ ❢♦r ❛❧❧ ❛♥❞ ✈❛❧✐❞ tr✐♣❧❡s ✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✸ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❚r❡❡ ♣❡rt✉r❜❛t✐♦♥s

❉❡✜♥❡ ❛ tr❡❡ ♣❡rt✉r❜❛t✐♦♥ ❜② Sjk

i

T = T − ij + ik. P✐❝t✉r❡ ✇✐❧❧ ❜❡ ❤❡r❡ ❲❡ s❛② F : Tn → R ✐s α✲▲✐♣s❝❤✐t③ ♦♥ T ⊆ Tn ✐❢ |F (T ) − F (Sjk

i

T )| ≤ α ❢♦r ❛❧❧ T ∈ T ❛♥❞ ✈❛❧✐❞ tr✐♣❧❡s (i, j, k)✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✸ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❆❧♠♦st s✉♣❡r♣♦s❛❜❧❡ ♣❡rt✉r❜❛t✐♦♥s

❆t ❛ ❜✐❣ ❞✐st❛♥❝❡✱ ✇❡ ✇❛♥t t❤❛t

F (Sjk

i

Sbc

a T ) − F (T ) ≈

• F (Sjk

i

T ) − F (T )

• +
• F (Sbc

a T ) − F (T )

• .

P✐❝t✉r❡ ✇✐❧❧ ❜❡ ❤❡r❡ ❋♦r ✱ ✇❡ s❛② ✐s ✲s✉♣❡r♣♦s❛❜❧❡ ♦♥ ✐❢ ❢♦r ❛❧❧ ❛♥❞ ✈❛❧✐❞ tr✐♣❧❡s ✱ ✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✹ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❆❧♠♦st s✉♣❡r♣♦s❛❜❧❡ ♣❡rt✉r❜❛t✐♦♥s

❆t ❛ ❜✐❣ ❞✐st❛♥❝❡✱ ✇❡ ✇❛♥t t❤❛t

F (Sjk

i

Sbc

a T ) − F (T ) ≈

• F (Sjk

i

T ) − F (T )

• +
• F (Sbc

a T ) − F (T )

• .

P✐❝t✉r❡ ✇✐❧❧ ❜❡ ❤❡r❡ ❋♦r ρ : N → R+✱ ✇❡ s❛② F : Tn → R ✐s ρ✲s✉♣❡r♣♦s❛❜❧❡ ♦♥ T ⊆ Tn ✐❢

|F (T ) − F (Sjk

i

T ) − F (Sbc

a T ) + F (Sjk i

Sbc

a T )| ≤ ρ(dT ({j, k}, {b, c})).

❢♦r ❛❧❧ T ∈ T ❛♥❞ ✈❛❧✐❞ tr✐♣❧❡s (i, j, k)✱ (a, b, c)✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✹ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

▼❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠ ✭■✳✱ ❙♦✉t❤✇❡❧❧✱ ❩❤✉❦♦✈s❦✐✐✮ ❙✉♣♣♦s❡ F : Tn → R ✐s α✲▲✐♣s❝❤✐t③ ♥❞ ρ✲s✉♣❡r♣♦s❛❜❧❡ ♦♥ t❤❡ s❡t ♦❢ tr❡❡s ✇✐t❤ ❞❡❣r❡❡s ❛t ♠♦st log n✳ ■❢ t❤❡r❡ ✐s ε > 0 s✉❝❤ t❤❛t nα3 σ3 + n1/4α σ + n1/4 n

d=1 dρ(d)

σ = O(n−ε) ❛♥❞ |F (T ) − µ| = eO(log n)σ t❤❡♥ F (T) ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧✳ ▼♦r❡♦✈❡r✱ δK(F ) = O(n−ε′) ❢♦r ❛♥② ε′ ∈ (0, ε)✳ ❍❡r❡✱ ❛s ❜❡❢♦r❡✱ µ = EF (T) ❛♥❞ σ2 = Var F (T)✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✺ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❱❡rt✐❝❡s ♦❢ ❣✐✈❡♥ ❞❡❣r❡❡

❲❡ r❡❝❛❧❧ t❤❛t✱ s❡❡ ❬▼♦♦♥✱ ❈♦✉♥t✐♥❣ ▲❛❜❡❧❧❡❞ tr❡❡s✱✶✾✼✵❪✱ ENk(T) = npk + O(1), Var Nk(T) = npk(1 − pk) − np2

k(k − 2)2 + O(1),

✇❤❡r❡ pk = (e(k − 1)!)−1✳ ❈♦r♦❧❧❛r② ✶ Nk(T) ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ❛♥❞ δK (Nk) = O

• n−1/4+ǫ

. Pr♦♦❢✳ ❍❡r❡✱ ✇❡ ❤❛✈❡ α = 2 s✐♥❝❡ T ❛♥❞ Sjk

i

T ❤❛✈❡ t❤❡ s❛♠❡ ❞❡❣r❡❡s ❡①❝❡♣t ❢♦r j ❛♥❞ k✳ ❲❡ ❝❛♥ ♣✉t ρ(d) = 0 ❢♦r d ≥ 1✳ ❆♣♣❧② ❈▲❚ ❢♦r tr❡❡s✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✻ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❖❝❝✉rr❡♥❝❡s ♦❢ ❛ tr❡❡ ♣❛tt❡r♥

❚❤❡♦r❡♠ ✭❈❤②③❛❦✱ ❉r♠♦t❛✱ ❑❧❛✉s♥❡r✱ ❑♦❦✱ ✷✵✵✽✮ ▲❡t M ❜❡ ❛ ❣✐✈❡♥ tr❡❡✳ ❚❤❡♥✱ t❤❡ ♥✉♠❜❡r ♦❢ ♦❝❝✉rr❡♥❝❡s ♦❢ M ✐♥ T ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ✇✐t❤ ♠❡❛♥ ❛♥❞ ✈❛r✐❛♥❝❡ ❛s②♠♣t♦t✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ µn ❛♥❞ σ2n✱ r❡s♣❡❝t✐✈❡❧②✱ ✇❤❡r❡ µ > 0 ❛♥❞ σ2 > 0 ❞❡♣❡♥❞ ♦♥ t❤❡ ♣❛tt❡r♥ M ❛♥❞ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❡①♣❧✐❝✐t❧② ❛♥❞ ❛❧❣♦r✐t❤♠✐❝❛❧❧② ❛♥❞ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ♣♦❧②♥♦♠✐❛❧s ✭✇✐t❤ r❛t✐♦♥❛❧ ❝♦❡✣❝✐❡♥ts✮ ✐♥ 1/e✳ P✐❝t✉r❡ ✇✐❧❧ ❜❡ ❤❡r❡

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✼ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❖❝❝✉rr❡♥❝❡s ♦❢ ❛ tr❡❡ ♣❛tt❡r♥

❚❤❡♦r❡♠ ✭❈❤②③❛❦✱ ❉r♠♦t❛✱ ❑❧❛✉s♥❡r✱ ❑♦❦✱ ✷✵✵✽✮ ▲❡t M ❜❡ ❛ ❣✐✈❡♥ tr❡❡✳ ❚❤❡♥✱ t❤❡ ♥✉♠❜❡r ♦❢ ♦❝❝✉rr❡♥❝❡s ♦❢ M ✐♥ T ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ✇✐t❤ ♠❡❛♥ ❛♥❞ ✈❛r✐❛♥❝❡ ❛s②♠♣t♦t✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ µn ❛♥❞ σ2n✱ r❡s♣❡❝t✐✈❡❧②✱ ✇❤❡r❡ µ > 0 ❛♥❞ σ2 > 0 ❞❡♣❡♥❞ ♦♥ t❤❡ ♣❛tt❡r♥ M ❛♥❞ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❡①♣❧✐❝✐t❧② ❛♥❞ ❛❧❣♦r✐t❤♠✐❝❛❧❧② ❛♥❞ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ♣♦❧②♥♦♠✐❛❧s ✭✇✐t❤ r❛t✐♦♥❛❧ ❝♦❡✣❝✐❡♥ts✮ ✐♥ 1/e✳ P✐❝t✉r❡ ✇✐❧❧ ❜❡ ❤❡r❡

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✼ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❚✇♦✲♣❛t❤s

▲❡t P2(T ) ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ t✇♦✲♣❛t❤s✳ ❲❡ ❤❛✈❡ P2(T ) = n

i=1

di

2

• ✳

❈♦r♦❧❧❛r② ✷ P2(T) ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ❛♥❞ δK (P2) = O

• n−1/4+ǫ

. Pr♦♦❢✳ ■t ✐s ❛ r♦✉t✐♥❡ t♦ ❝❛❧❝✉❧❛t❡ t❤❛t ❋♦r ❛ tr❡❡ ✇❤✐❝❤ ❞❡❣r❡❡s ❛t ♠♦st ✱ ✇❡ ❤❛✈❡

❲❡ ❝❛♥ ❛❧s♦ ♣✉t ❢♦r ✳ ❆♣♣❧② ❈▲❚ ❢♦r tr❡❡s✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✽ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❚✇♦✲♣❛t❤s

▲❡t P2(T ) ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ t✇♦✲♣❛t❤s✳ ❲❡ ❤❛✈❡ P2(T ) = n

i=1

di

2

• ✳

❈♦r♦❧❧❛r② ✷ P2(T) ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ❛♥❞ δK (P2) = O

• n−1/4+ǫ

. Pr♦♦❢✳ ■t ✐s ❛ r♦✉t✐♥❡ t♦ ❝❛❧❝✉❧❛t❡ t❤❛t EP2(Tn) ∼ 3n/2, Var P2(Tn) ∼ n/2. ❋♦r ❛ tr❡❡ T ✇❤✐❝❤ ❞❡❣r❡❡s d1, . . . , dn ❛t ♠♦st log n✱ ✇❡ ❤❛✈❡

|P2(T ) − P2(Sjk

i

T )| ≤

• dj

2

• +
• dk

2

• −
• dj − 1

2

• +
• dk + 1

2

• ≤ log n.

❲❡ ❝❛♥ ❛❧s♦ ♣✉t ρ(d) = 0 ❢♦r d ≥ 1✳ ❆♣♣❧② ❈▲❚ ❢♦r tr❡❡s✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✽ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❚❤r❡❡✲♣❛t❤s

❍❡r❡✱ ✇❡ ❤❛✈❡ P3(T ) =

ij∈E(T ) 1 2(di − 1)(dj − 1).

❈♦r♦❧❧❛r② ✷ P3(T) ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ❛♥❞ δK (P3) = O

• n−1/4+ǫ

. Pr♦♦❢✳ ■t ✐s ❛ r♦✉t✐♥❡ t♦ ❝❛❧❝✉❧❛t❡ t❤❛t ❋♦r ❛ tr❡❡ ✇❤✐❝❤ ❞❡❣r❡❡s ❛t ♠♦st ✱ ✇❡ ❤❛✈❡

❲❡ ❝❛♥ ♣✉t ❛♥❞ ❢♦r ✳ ❆♣♣❧② ❈▲❚ ❢♦r tr❡❡s✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✾ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❚❤r❡❡✲♣❛t❤s

❍❡r❡✱ ✇❡ ❤❛✈❡ P3(T ) =

ij∈E(T ) 1 2(di − 1)(dj − 1).

❈♦r♦❧❧❛r② ✷ P3(T) ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ❛♥❞ δK (P3) = O

• n−1/4+ǫ

. Pr♦♦❢✳ ■t ✐s ❛ r♦✉t✐♥❡ t♦ ❝❛❧❝✉❧❛t❡ t❤❛t EP3(Tn) ∼ 2n, Var P3(Tn) ∼ 3n. ❋♦r ❛ tr❡❡ T ✇❤✐❝❤ ❞❡❣r❡❡s d1, . . . , dn ❛t ♠♦st log n✱ ✇❡ ❤❛✈❡

|P3(T ) − P3(Sjk

i

T )| = O(log2 n). ❲❡ ❝❛♥ ♣✉t ρ(1) = 2α ❛♥❞ ρ(d) = 0 ❢♦r d ≥ 2✳ ❆♣♣❧② ❈▲❚ ❢♦r tr❡❡s✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✾ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❈❉❋s ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ t✇♦✲♣❛t❤s ✐♥ T ∈ Tn

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✶✵ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❈❉❋s ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ t❤r❡❡✲♣❛t❤s ✐♥ T ∈ Tn

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✶✶ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❚r❡❡ ❛✉t♦♠♦r♣❤✐s♠s

❨✉ ✭✷✵✶✷✮✿ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ♦❢ E log | Aut(T)| ❛♥❞ ❝♦♥❝❡♥tr❛t✐♦♥ ♣r♦♣❡rt②✳ ▲♦❣♥♦r♠❛❧ ❧✐♠✐t ❧❛✇ ✐s ❝♦♥❥❡❝t✉r❡❞✳ ❲❛❣♥❡r ✭❇❛♥✛ t❛❧❦✱ ❖❝t♦❜❡r ✷✵✶✻✮✿ log | Aut(T)| ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ✇✐t❤ µ ≈ 0.052290n ❛♥❞ σ2 ≈ 0.039498n✳ ❚♦ ❛♣♣❧② ♦✉r r❡s✉❧t✱ ✇❡ ♦♥❧② ♥❡❡❞ t❤❛t σ2 ✐s ❧✐♥❡❛r ❛♥❞ Aut(T) ❝❛♥ ❜❡ r❡str✐❝t❡❞ t♦ t❤❡ s✉❜❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② ♣❡r♠✉t❛t✐♦♥s ♦❢ s♠❛❧❧ ❜r❛♥❝❤❡s✳ ❈♦r♦❧❧❛r② ✸ log | Aut(T)| ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ❛♥❞ δK (log | Aut |) = O

• n−1/4+ǫ

. Pr♦♦❢✳ ■❢ t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛ ✈❡rt❡① ✐♥ t❤❡♥ ✳ ❚❤✉s✱ ❲❡ ❝❛♥ ♣✉t ❢♦r ❛♥❞ ✱ ♦t❤❡r✇✐s❡ ✭t❤✐s ✇♦r❦s ❢♦r ♣❡r♠✉t❛t✐♦♥s ♦❢ s♠❛❧❧ ❜r❛♥❝❤❡s✮✳ ❆♣♣❧② ❈▲❚ ❢♦r tr❡❡s✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✶✷ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❚r❡❡ ❛✉t♦♠♦r♣❤✐s♠s

❚♦ ❛♣♣❧② ♦✉r r❡s✉❧t✱ ✇❡ ♦♥❧② ♥❡❡❞ t❤❛t σ2 ✐s ❧✐♥❡❛r ❛♥❞ Aut(T) ❝❛♥ ❜❡ r❡str✐❝t❡❞ t♦ t❤❡ s✉❜❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② ♣❡r♠✉t❛t✐♦♥s ♦❢ s♠❛❧❧ ❜r❛♥❝❤❡s✳ ❈♦r♦❧❧❛r② ✸ log | Aut(T)| ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ❛♥❞ δK (log | Aut |) = O

• n−1/4+ǫ

. Pr♦♦❢✳ ■❢ t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛ ✈❡rt❡① ✐♥ t❤❡♥ ✳ ❚❤✉s✱ ❲❡ ❝❛♥ ♣✉t ❢♦r ❛♥❞ ✱ ♦t❤❡r✇✐s❡ ✭t❤✐s ✇♦r❦s ❢♦r ♣❡r♠✉t❛t✐♦♥s ♦❢ s♠❛❧❧ ❜r❛♥❝❤❡s✮✳ ❆♣♣❧② ❈▲❚ ❢♦r tr❡❡s✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✶✷ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❚r❡❡ ❛✉t♦♠♦r♣❤✐s♠s

❚♦ ❛♣♣❧② ♦✉r r❡s✉❧t✱ ✇❡ ♦♥❧② ♥❡❡❞ t❤❛t σ2 ✐s ❧✐♥❡❛r ❛♥❞ Aut(T) ❝❛♥ ❜❡ r❡str✐❝t❡❞ t♦ t❤❡ s✉❜❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② ♣❡r♠✉t❛t✐♦♥s ♦❢ s♠❛❧❧ ❜r❛♥❝❤❡s✳ ❈♦r♦❧❧❛r② ✸ log | Aut(T)| ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ❛♥❞ δK (log | Aut |) = O

• n−1/4+ǫ

. Pr♦♦❢✳ ■❢ S t❤❡ st❛❜✐❧✐③❡r ♦❢ ❛ ✈❡rt❡① ✐♥ Aut(T ) t❤❡♥ |S| ≥ | Aut(T )|/n✳ ❚❤✉s✱

• log | Aut T | − log | Aut(Sjk

i

T )|

• = O(log n).

❲❡ ❝❛♥ ♣✉t ρ(d) = 2α ❢♦r d ≤ log n ❛♥❞ ρ(d) = 0✱ ♦t❤❡r✇✐s❡ ✭t❤✐s ✇♦r❦s ❢♦r ♣❡r♠✉t❛t✐♦♥s ♦❢ s♠❛❧❧ ❜r❛♥❝❤❡s✮✳ ❆♣♣❧② ❈▲❚ ❢♦r tr❡❡s✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✶✷ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

▼❛rt✐♥❣❛❧❡s

▲❡t P = (Ω, F, P ) ❜❡ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❆ s❡q✉❡♥❝❡ Y0, . . . , Ym ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦♥ P ✐s ❝❛❧❧❡❞ ♠❛rt✐♥❣❛❧❡ ✐❢ ✐t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❧❛t✐♦♥s✿ ✶✳ ❊❛❝❤ Yi ✐s ❛♥ ✐♥t❡❣r❛❜❧❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇❤✐❝❤ ✐s ♠❡❛s✉r❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜ σ✲✜❡❧❞ Fi ⊆ F✳ ✷✳ ❚❤❡ σ✲✜❡❧❞s Fi ❢♦r♠ ❛ ✜❧t❡r✱ ✐✳❡✳ F0 ⊆ · · · ⊆ Fm✳ ✸✳ ❋♦r ❡✈❡r② 1 ≤ i ≤ m✱ ✇❡ ❤❛✈❡ E(Yi | Fi−1) = Yi−1 ❛✳s✳ ❊①❛♠♣❧❡✿ ❉♦♦❜ ♠❛rt✐♥❣❛❧❡ ❢♦r f(X1, . . . , Xm)✿ ❢♦r ω = (x1, . . . , xm) ∈ Ω✱ Yi(ω) = fi(x1, . . . , xi) = E(f(X1, . . . , Xm) | X1 = x1, . . . , Xi = xi) ✭✐♥ ❝❛s❡ ♦❢ ✐♥❞❡♣❡♥❞❡♥❝❡✮ = Ef(x1, . . . , xi, Xi+1, . . . , Xm).

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✶✸ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

▼❛rt✐♥❣❛❧❡ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠

▲❡t Y0, . . . , Ym ❜❡ ❛ ♠❛rt✐♥❣❛❧❡ ✇✐t❤ r❡s♣❡❝t t♦ ❛ ✜❧t❡r F0, . . . , Fm ✇✐t❤ F0 = {∅, Ω}✳ ❙✉♣♣♦s❡ t❤❛t✱ ❢♦r s♦♠❡ α > 0✱ ❛✳s✳ |Yi − Yi−1| ≤ α, i = 1, . . . , m, m

i=1 Var(Yi | Fi−1)

Var Ym

p

− → 1 ❛s m → ∞, ❛♥❞ ❛ ▲✐♥❞❡❜❡r❣✲t②♣❡ ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞✳ ❚❤❡♥✱ ✐t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t Ym − EYm (Var Ym)1/2

d

− → N(0, 1).

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✶✹ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❙t❡♣s t♦ ♥♦r♠❛❧✐t②

✭✶✮ ❘❡♣r❡s❡♥t F (T) ❛s f(X1, . . . , Xm)✳ ❚❛❦❡ t❤❡ ❉♦♦❜ ♠❛rt✐♥❣❛❧❡✳ ✭✷✮ ❇♦✉♥❞ t❤❡ ♠❛rt✐♥❣❛❧❡ ❞✐✛❡r❡♥❝❡s |Yi − Yi−1|✳ ✭✸✮ ❆❧s♦ ❜♦✉♥❞ | Var(Ym | Fi) − E(Var(Ym | Fi) | Fi−1)|✳ ✭✹✮ ❈♦♠❜✐♥❡ ✭✸✮ ❛♥❞ ❆③✉♠❛✬s ✐♥❡q✉❛❧✐t②✳ ❆♣♣❧② t❤❡ ♠❛rt✐♥❣❛❧❡ ❈▲❚✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✶✺ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

▼❛rt✐♥❣❛❧❡ ❝♦♥str✉❝t✐♦♥

❲❡ ✉s❡ ❆❧❞♦✉s✲❇r♦❞❡r ❛❧❣♦rt✐❤♠ ❢♦r ❣❡♥❡r❛t✐♦♥ ♦❢ s♣❛♥♥✐♥❣ tr❡❡s ♦❢ ❛ ❣r❛♣❤ G✳ ❲❤❡♥ G = Kn✱ n ≥ 2 ✐t ❝❛♥ ❜❡ r❡♣❤r❛s❡❞ ❛s ❢♦❧❧♦✇s✿ ■✳ ❋♦r 1 ≤ i ≤ n − 1 ❝♦♥♥❡❝t ✈❡rt❡① i + 1 t♦ ✈❡rt❡① Vi = min{i, Ui}✱ ✇❤❡r❡ U = (U1, . . . , Un−1) ✐s ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ♦♥ [n]n−1✳ ■■✳ ❘❡❧❛❜❡❧ ✈❡rt✐❝❡s 1, . . . , n ❛s X1, . . . , Xn✱ ✇❤❡r❡ X = (X1, . . . , Xn) ✐s ❛ ✉♥✐❢♦r♠ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢r♦♠ Sn✳ ❚❤✉s✱ ✇❡ r❡♣r❡s❡♥t F (T) ❛s f(U, X)✳ ❚❤❡♥✱ ✇❡ ❥✉st ❢♦❧❧♦✇ t❤❡ st❡♣s ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s s❧✐❞❡✳

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✶✻ ✴ ✶✼

• ❡♥❡r❛❧ t❤❡♦r❡♠

❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✶✼ ✴ ✶✼