An asymptotic distribution theory for Eulerian recurrences with applications

Abstract

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.

Probabilistic consequences of some polynomial recurrences

“…Theorem 2 is one such criteria for Gaussian limits, and [13, Proposition 11] is one such criteria for Poisson limits. We refer to [15] for recent results in this direction. It would also be interesting to have criteria for convergence to other distributions like Rayleigh or, more generally, a Weilbull distribution that is, a distribution with density px p−1 e −x p , x > 0, for a fixed p > 0; Rayleigh distribution corresponds to p = 2.…”

“…We adapt a result of [17] to do this in our particular situation which may be useful in other settings. In addition, this paper discusses similar types of polynomial recurrences which are found in the literature (see [1,6,12,15]). This paper establishes a framework to approach such recurrences and derive the probabilistic consequences of them.…”

“…(iii) Theorem 2 can be used to prove asymptotic normality for other recurrences, including HJ, DHH mentioned in this paper and many others; see a recent preprint [15] for subsequent developments.…”

“…In all these examples, the polynomials are associated with integer valued random variables and as we have done in Section 2, it is of interest to understand what bearing the form of a recurrence has on the probabilistic properties of these random variables. This is, of course, not a new idea and in various forms has been studied for a long time (see, eg, many results and references in [8] and in a recent preprint [15]). Still, we believe that there is more work to be done to better understand the probabilistic consequences of the above recurrences and would like to establish a general framework for such work.…”

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

“…(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 (x) = n (x) + (x), and…”