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❚❤❡ ❞✐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 T n t❤❡ s❡t ♦❢ ❛❧❧ ❧❛❜❡❧❧❡❞ tr❡❡s ✇✐t❤ ✈❡rt✐❝❡s { 1 , . . . , n } ✳ ▲❡t T ❜❡ ❛ ✉♥✐❢♦r♠ r❛♥❞♦♠ ❡❧❡♠❡♥t ♦❢ T n ✳ ■❢ F : T n → R ✐s ✏♥✐❝❡✑ t❤❡♥ F ( T ) ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧✱ ✐✳❡✳ � t � F ( T ) − µ � n → ∞ e − x 2 / 2 dx, 1 Pr ≤ t − → Φ( t ) = √ σ 2 π −∞ ✇❤❡r❡ µ = E F ( T ) ❛♥❞ σ 2 = Var F ( T ) . � F ( T ) − µ � � � ❲❡ ❛♥❛❧②③❡ ❛❧s♦ t❤❡ r❛t❡s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ δ K ( F ) = � CDF − Φ ∞ t♦ ③❡r♦✳ � � σ � ▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✷ ✴ ✶✼

❲❡ s❛② ✐s ✲▲✐♣s❝❤✐t③ ♦♥ ✐❢ ❢♦r ❛❧❧ ❛♥❞ ✈❛❧✐❞ tr✐♣❧❡s ✳ ●❡♥❡r❛❧ t❤❡♦r❡♠ ❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s ❚r❡❡ ♣❡rt✉r❜❛t✐♦♥s ❉❡✜♥❡ ❛ tr❡❡ ♣❡rt✉r❜❛t✐♦♥ ❜② S jk T = T − ij + ik. i P✐❝t✉r❡ ✇✐❧❧ ❜❡ ❤❡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 S jk T = T − ij + ik. i P✐❝t✉r❡ ✇✐❧❧ ❜❡ ❤❡r❡ ❲❡ s❛② F : T n → R ✐s α ✲▲✐♣s❝❤✐t③ ♦♥ T ⊆ T n ✐❢ | F ( T ) − F (S jk T ) | ≤ α i ❢♦r ❛❧❧ T ∈ T ❛♥❞ ✈❛❧✐❞ tr✐♣❧❡s ( i, j, k ) ✳ ▼✐❦❤❛✐❧ ■s❛❡✈ ✭▼♦♥❛s❤ ❯♥✐✈❡rs✐t②✮ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ tr❡❡ ♣❛r❛♠❡t❡rs ❆♣r✐❧ ✷✾✱ ✷✵✶✾ ✸ ✴ ✶✼

❋♦r ✱ ✇❡ s❛② ✐s ✲s✉♣❡r♣♦s❛❜❧❡ ♦♥ ✐❢ ❢♦r ❛❧❧ ❛♥❞ ✈❛❧✐❞ tr✐♣❧❡s ✱ ✳ ●❡♥❡r❛❧ t❤❡♦r❡♠ ❊①❛♠♣❧❡s ▼❛rt✐♥❣❛❧❡s ❆❧♠♦st s✉♣❡r♣♦s❛❜❧❡ ♣❡rt✉r❜❛t✐♦♥s ❆t ❛ ❜✐❣ ❞✐st❛♥❝❡✱ ✇❡ ✇❛♥t t❤❛t � � � � F (S jk S bc F (S jk F (S bc a T ) − F ( T ) ≈ T ) − F ( T ) + a T ) − F ( T ) . i i P✐❝t✉r❡ ✇✐❧❧ ❜❡ ❤❡r❡ ▼✐❦❤❛✐❧ ■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 (S jk S bc F (S jk F (S bc a T ) − F ( T ) ≈ T ) − F ( T ) + a T ) − F ( T ) . i i P✐❝t✉r❡ ✇✐❧❧ ❜❡ ❤❡r❡ ❋♦r ρ : N → R + ✱ ✇❡ s❛② F : T n → R ✐s ρ ✲s✉♣❡r♣♦s❛❜❧❡ ♦♥ T ⊆ T n ✐❢ | F ( T ) − F (S jk T ) − F (S bc a T ) + F (S jk S bc a T ) | ≤ ρ ( d T ( { j, k } , { b, c } )) . i i ❢♦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 : T n → 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 + n 1 / 4 � n nα 3 σ 3 + n 1 / 4 α d =1 dρ ( d ) = O ( n − ε ) σ σ ❛♥❞ | F ( T ) − µ | = e O (log n ) σ t❤❡♥ F ( T ) ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧✳ ▼♦r❡♦✈❡r✱ δ K ( F ) = O ( n − ε ′ ) ❢♦r ❛♥② ε ′ ∈ (0 , ε ) ✳ ❍❡r❡✱ ❛s ❜❡❢♦r❡✱ µ = E F ( T ) ❛♥❞ σ 2 = 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 ✱✶✾✼✵❪✱ E N k ( T ) = np k + O (1) , k ( k − 2) 2 + O (1) , Var N k ( T ) = np k (1 − p k ) − np 2 ✇❤❡r❡ p k = ( e ( k − 1)!) − 1 ✳ ❈♦r♦❧❧❛r② ✶ n − 1 / 4+ ǫ � � N k ( T ) ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ❛♥❞ δ K ( N k ) = O . Pr♦♦❢✳ ❍❡r❡✱ ✇❡ ❤❛✈❡ α = 2 s✐♥❝❡ T ❛♥❞ S jk T ❤❛✈❡ t❤❡ s❛♠❡ ❞❡❣r❡❡s ❡①❝❡♣t i ❢♦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✐❧ ✷✾✱ ✷✵✶✾ ✻ ✴ ✶✼

P✐❝t✉r❡ ✇✐❧❧ ❜❡ ❤❡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 ❛♥❞ σ 2 n ✱ r❡s♣❡❝t✐✈❡❧②✱ ✇❤❡r❡ µ > 0 ❛♥❞ σ 2 > 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 ✳ ▼✐❦❤❛✐❧ ■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 ❛♥❞ σ 2 n ✱ r❡s♣❡❝t✐✈❡❧②✱ ✇❤❡r❡ µ > 0 ❛♥❞ σ 2 > 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✐❧ ✷✾✱ ✷✵✶✾ ✼ ✴ ✶✼