On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey--Wilson, Al-Salam--Chihara and q-ultraspherical polynomials

Abstract

We study properties of compactly supported, 4 parameter \newline $(\rho _{12},\rho _{23},\rho _{13},q)\in (-1,1)^{\times 4}$ family of continuous type 3 dimensional distributions, that have the property that for $q\rightarrow 1^{-}$ this family tends to some 3 dimensional Normal distribution. For $q=0$ we deal with 3 dimensional generalization of Kesten--McKay distribution. In a very special case when $\rho _{12}\rho _{13}\rho _{23}=q$ all one dimensional marginals are identical, semicircle distributions. We find both all marginal as well as all conditional distributions. Moreover, we find also families of polynomials that are orthogonalized by these one-dimensional margins and one-dimensional conditional distributions. Consequently, we find moments of both conditional and unconditional distributions of dimensions one and two. In particular, we show that all one-dimensional and two-dimensional conditional moments of, say, order $n$ and are polynomials of the same order $n$ in the conditioning random variables. Finding above mentioned orthogonal polynomials leads us to a probabilistic interpretation of these polynomials. Among them are the famous Askey-Wilson, Al-Salam--Chihara polynomials considered in the complex, but conjugate, parameters, as well as q-Hermite and Rogers polynomials. It seems that this paper is one of the first papers that give a probabilistic interpretation of Rogers (continuous q-ultraspherical) polynomials.