Generalized Erd os-Tur an laws for the order of random permutation - PowerPoint PPT Presentation

Generalized Erd˝ os-Tur´ an laws for the order of random permutation Alexander Gnedin (QMUL, London) Alexander Iksanov (Kiev University) Alexander Marynych (Kiev University)

Order of permutation σ 12 = id σ = (1 9 6 2)(3 7 5)(4 8) , l . c . m . (4 , 3 , 2) = 12 ◮ For permutation of [ n ] := { 1 , 2 , . . . , n } K n , r := # cycles of length r , ( K n , r ; r ∈ [ n ]) cycle partition ◮ O n := l . c . m . { r : K n , r > 0 } .

Erd˝ os-Tur´ an laws ◮ Erd˝ os-Tur´ an (1967): For uniformly random permutation of [ n ] 2 log 2 n log O n − 1 d → N (0 , 1) � 3 log 3 n 1

Ewens’ permutations Ewens’ distribution on permutations of [ n ] θ K n P ( σ ) = θ ( θ + 1) . . . ( θ + n − 1) , θ > 0 � K n := K n , r # of cycles r The distribution of ( K n , r ; r ∈ [ n ]) is the Ewens sampling formula. ◮ Arratia and Tavar´ e 1992: For Ewens’ permutation of [ n ] 2 log 2 n log O n − θ d � → N (0 , 1) 3 log 3 n θ ◮ log O n approximable by log T n = � r log r K n , r ◮ K n , r ’s asymptotically independent, Poisson( θ/ r )

Poisson-Dirichlet/GEM connection d P ( W ∈ dx ) = θ x θ − 1 dx , x ∈ (0 , 1) W = Beta ( θ, 1) , W 1 , W 2 , . . . i.i.d. copies of W ◮ PD/GEM random discrete distribution P j = W 1 · · · W j − 1 (1 − W j ) , j ∈ N ◮ For sample of size n from ( P j ), K n , r is the number of values j ∈ N represented r times (so � r rK n , r = n ) ◮ From random partition to permutation: conditionally on the cycle partition ( K n , r ; r ∈ [ n ]) the permutation is uniformly distributed. ◮ LLN: P j ’s are asymptotic frequencies of ‘big’ components of the partition

General stick-breaking factor W d ◮ P j = W 1 · · · W j − 1 (1 − W j ) , j ∈ N , with i.i.d. W j = W , where W is a ‘stick-breaking factor’ with general distribution on [0 , 1] ◮ generate partition/permutation of [ n ] by sampling n elements from ( P j ) and letting K n , r to be the number of integer values represented r times in the sample. ◮ Problem: What is the limit distribution of log O n − b n a n for suitable centering/scaling constants b n , a n ?

Permutations with distribution of the Gibbs form k � p ( λ 1 , . . . , λ k ) = c n , k θ λ i i =1 (Betz/Ueltschi/Velenik, Nikeghbali/Zeindler, . . . ) are not permutations derived by the stick-breaking, unless they belong to Ewens’s family. ◮ Regenerative property: the collection of cycle-sizes coincides with the set of jumps of a decreasing Markov chain with transition matrix � n � E [ W m (1 − W ) n − m ] q ( n , m ) = , 0 ≤ m ≤ n − 1 . m 1 − E W n starting state n and absorbing state 0. Example: for Ewens’ permutations � n � ( θ ) n − m m ! q ( n , m ) = . m n ( θ + 1) n − 1

For Ewens’ permutations, general separable (additive) functionals � h ( r ) K n , r r have been studied by Babu and Manstavicius (2002, 2009) for unbounded functions h (we need h ( r ) = log r ). For the permutations derived from stick-breaking: ◮ K n , r ’s are not asymptotically independent, ◮ K n , r ’s converge (if E | log W | < ∞ ) to some multivariate discrete distribution, which is intractable (G.,Iksanov and Roesler 2008)

◮ Density assumption P ( W ∈ dx ) = f ( x ) dx , x ∈ (0 , 1), ◮ Define µ := E | log W | , σ 2 := Var (log W ) , ν := E | log(1 − W ) | ; we shall assume µ < ∞ , σ 2 ≤ ∞ , ν ≤ ∞ .

Normal limit I Suppose x β (1 − x ) β f ( x ) < ∞ for some β ∈ [0 , 1) . ( I ) : sup x ∈ [0 , 1] Then log O n − b n d → N (0 , 1) , a n with constants b n = log 2 n 2 µ � σ 2 log 3 n a n = 3 µ 3 Example: f = Beta( θ, ζ ); θ, ζ > 0.

Normal limit IIa (II) : Suppose (for some small δ ) f is nonincreasing in [0 , δ ] , nondecreasing in [1 − δ, 1] and bounded on [ δ, 1 − δ ] . If σ 2 < ∞ then log O n − b n d → N (0 , 1) , a n for � � � log 2 n � z log 2 n b n = 1 − P (log | 1 − W | > x ) dxdz µ 2 0 0 � σ 2 log 3 n a n = . 3 µ 3

Normal limit IIb If σ 2 = ∞ and � x y 2 P ( | log W | ∈ dy ) ∼ ℓ ( x ) 0 for function ℓ of slow variation at ∞ , then the normal limit holds with � c ⌊ log n ⌋ log n a n = , 3 µ 3 where c n is any sequence satisfying n ℓ ( c n ) → 1 . c 2 n

Stable limit IIc If for some α ∈ (1 , 2) and ℓ of slow variation at ∞ P ( | log W | > x ) ∼ x − α ℓ ( x ) , then the limit is α -stable with characteristic function � � � � cos πα 2 + i sin πα −| u | α Γ(1 − α ) u �→ exp sgn u . 2 The centering b n is as in IIa and scaling c ⌊ log n ⌋ log n a n = (( α + 1) µ α +1 ) 1 /α

Reduction to T n For T n = � n r =1 r K n , r E | log O n − log T n | = O (log n log log n ) , under any of the assumptions I, IIa, IIb, IIc.

Perturbed random walk ξ > 0 , η ≥ 0 any dependent random variables, ( ξ j , η j ) i.i.d. copies of ( ξ, η ) S k = ξ 1 + · · · + ξ k ◮ Perturbed random walk � S k = S k − 1 + η k ◮ For ξ = − log W , η = − log(1 − W ) , the log-frequencies (log P k , k ≥ 1) is a perturbed RW ◮ Number of ‘renewals’ N ( x ) := # { k ≥ 1 : � � N ( x ) := # { k ≥ 0 : S k ≤ x } , S k ≤ x }

� x ϕ ( x ) := P ( η > y ) dy 0 Assume that µ = E ξ < ∞ and for some c ( x ) N ( x ) − x µ d → Z , as x → ∞ . c ( x ) Then Z is a stable random variable (Bingham 1973), and � � � x N ( y ) − y − ϕ ( y ) � 1 dy 0 µ d → Z ( y ) dy , as x → ∞ , xc ( x ) 0 where ( Z ( t ) , t ≥ 0) is a stable L´ evy process corresponding to Z d = Z (1) .

Open problem: generalize to the Ewens-Pitman two parameter family of random permutations.