Noncentral Limit Theorems and Appell Polynomials

Limit Theorems for Bivariate Appell

Consider the stationary linear process X t = P 1 u=?1 a(t?u) u , t 2 Z, where f u g is an i.i.d. nite variance sequence. The spectral density of fX t g may diverge at the origin (long-range dependence) or at any other frequency. Consider now the quadratic form Q N = P N t;s=1 b(t ? s)P m;n (X t ; X s), where P n;m (X t ; X s) denotes a non-linear function (Appell polynomial). We provide general...

Appell Polynomials and Their Relatives

This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials when polynomials in noncommuting variables are considered. They also fit well into the framework of free probability. For the free Appell polyn...

On Multiple Appell Polynomials

In this paper, we first define the multiple Appell polynomials and find several equivalent conditions for this class of polynomials. Then we give a characterization theorem that if multiple Appell polynomials are also multiple orthogonal, then they are the multiple Hermite polynomials.

Turan’s Inequality for Appell Polynomials

Matrix approach to Appell polynomials

In the last years the Appell polynomials, named after Paul Émile Appell which introduced them in 1880 [1], have gained renewed interest and has been studied by several authors. In particular, some new characterizations of Appell polynomials themselves through new approaches have been considered. To quote some of them we mention, for instance, the novel approach developed in [5], which makes use...