Multidimensional $q$-Normal and Related Distributions - Markov Case

Abstract

We define and study distributions in $\mathbb{R}^d$ that we call $q$-Normal. For $q=1$ they are really multidimensional Normal, for $q$ in $(-1,1)$ they have densities, compact support and many properties that resemble properties of ordinary multidimensional Normal distribution. We also consider some generalizations of these distributions and indicate close relationship of these distributions to Askey-Wilson weight function i.e. weight with respect to which Askey-Wilson polynomials are orthogonal and prove some properties of this weight function. In particular we prove a generalization of Poisson-Mehler expansion formula