Probabilistic consequences of some polynomial recurrences

Hitczenko, Paweł; Lohss, Amanda

doi:10.1002/rsa.20820

Cited by 4 publications

(4 citation statements)

References 20 publications

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“…Also our conditions are very easy to check in all cases we will discuss. Finally, recurrences similar to ours have been studied in the literature; see for example [78,80,124,221] and the references therein.…”

Section: A Normal Limit Theoremmentioning

confidence: 60%

“…We obtain, by Theorem 1, the CLT N 3 4 n, 7 48 n for the coefficients. This CLT was proved in [124] by the real-rootedness approach. The reciprocal polynomial Q n (v) = v n+1 P n 1 v satisfies the simpler recurrence Q n ∈ E (1 + v)n, 1 + v; 1 , where the right-hand side differs from that of P n only by a factor v. By the techniques of Section 3.1, the EGF has the exact form…”

Section: Polynomials With Quadratic α(V)mentioning

confidence: 86%

“…When the left-hand side of the Eulerian recurrence (9) is replaced by P n (v) (together with some boundary conditions), the same method of moments still applies, as already described in [124]. Note that in such cases, the presence of the crucial factor 1 − v in Eulerian recurrences is not essential for the application of the method of moments.…”

Section: Pólya Urn Modelsmentioning

confidence: 99%

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An asymptotic distribution theory for Eulerian recurrences with applications

Hwang

Chern

Duh

2020

Advances in Applied Mathematics

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We study linear recurrences of Eulerian type of the formwith P 0 (v) given, where α(v), β(v) and γ(v) are in most cases polynomials of low degrees. We characterize the various limit laws of the coefficients of P n (v) for large n using the method of moments and analytic combinatorial tools under varying α(v), β(v) and γ(v), and apply our results to more than two hundred of concrete examples when β(v) = 0 and more than three hundred when β(v) = 0 that we collected from the literature and from Sloane's OEIS database. The limit laws and the convergence rates we worked out are almost all new and include normal, half-normal, Rayleigh, beta, Poisson, negative binomial, Mittag-Leffler, Bernoulli, etc., showing the surprising richness and diversity of such a simple framework, as well as the power of the approaches used.

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Section: A Normal Limit Theoremmentioning

confidence: 60%

Section: Polynomials With Quadratic α(V)mentioning

confidence: 86%

Section: Pólya Urn Modelsmentioning

confidence: 99%

See 1 more Smart Citation

An asymptotic distribution theory for Eulerian recurrences with applications

Hwang

Chern

Duh

2020

Advances in Applied Mathematics

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“…For example, the real‐rootedness of the polynomials P n ( x ) was studied in 20 and in 12, under the assumption that g n ( x ) ≡ 0 in the latter paper. (Real‐rootedness can then be used to establish asymptotic normality of random variables that correspond to the polynomials { P n ( x )}; see for example 16 and references therein.) A detailed and vastly general study of the limiting properties of such random variables was carried out in 17 under the assumption that g n ≡ 0,

f_{n} false(x false) = n α false(x false) + γ false(x false)

, and

h_{n} false(x false) = false(1 prefix- x false) β false(x false)

, where

α

γ

, and

β

are typically polynomials of low degree.…”

Section: Discussionmentioning

confidence: 99%

Asymptotic normality of the number of corners in tableaux associated with the partially asymmetric simple exclusion process

Hitczenko

Yaroslavskiy

2020

Random Struct Algorithms

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In this paper, we study corners in tree-like and permutation tableaux. Tree-like tableaux are in bijection with other combinatorial structures, including permutation tableaux, and have a connection to the partially asymmetric simple exclusion process (PASEP), an important model of an interacting particles system. In particular, in the context of tree-like tableaux, a corner corresponds to a node occupied by a particle that could jump to the right while inner corners indicate a particle with an empty node to its left. Thus, the total number of corners represents the number of nodes at which PASEP can move, that is, the total current activity of the system. As the number of inner corners and regular corners is connected, we limit our discussion to just regular corners and show that asymptotically, the number of corners in a tableau of length n is normally distributed. Furthermore, since the number of corners in tree-like tableaux is closely related to the number of corners in permutation tableaux, we will discuss the corners in the context of the latter tableaux. Finally, using analogous techniques, we prove a central limit theorem for the number of corners in symmetric tree-like tableaux and type-B permutation tableaux.

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A Class of Polynomial Recurrences Resulting in $$({\text {n}}/{\text {log}}\,\,{\text {n}}, {\text {n}}/{\text {log}}^2{\text {n}})$$–Asymptotic Normality

Hitczenko

2024

La Matematica

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We consider sequences of polynomials that satisfy differential–difference recurrences. Polynomials satisfying such recurrences frequently appear as generating polynomials of integer valued random variables that are of interest in discrete mathematics. It is, therefore, of interest to understand the properties of such polynomials and their probabilistic consequences. We identify a class of polynomial recurrences that lead to a normal law with the expected value and the variance proportional to $$n/\log ~n$$ n / log n and $$n/\log ^2n$$ n / log 2 n , respectively. Examples include Stirling numbers of the second kind and other polynomials concerning set partitions as well as polynomials related to Whitney numbers of Dowling lattices.

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Probabilistic consequences of some polynomial recurrences

Cited by 4 publications

References 20 publications

An asymptotic distribution theory for Eulerian recurrences with applications

An asymptotic distribution theory for Eulerian recurrences with applications

Asymptotic normality of the number of corners in tableaux associated with the partially asymmetric simple exclusion process

A Class of Polynomial Recurrences Resulting in $$({\text {n}}/{\text {log}}\,\,{\text {n}}, {\text {n}}/{\text {log}}^2{\text {n}})$$–Asymptotic Normality

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