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

Paweł J. Szabłowski

Published 2017-12-12Version 1

We study properties of compactly supported 4 parameter $(\rho _{12},\rho _{23},\rho _{13},q)\in (-1,1)^{\times 4}$ family of continuous type 3 dimensional distributions, that have the property that for $q->1^{-}$ this family tends to some 3 dimensional Normal distribution. For $% q=0$ we deal with 3 dimensional generalization 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. More over we find also families of polynomials that are orthogonalized by these marginals and conditional distributions. Consequently we find moments both conditional and unconditional of such distributions. Among others we show that all conditional moments of say order $n$ are polynomials of the same order in conditioning random variables. In particular we give yet another probabilistic interpretation of the famous Askey-Wilson polynomials considered at complex but conjugate parameters as well as Rogers polynomials. It seems that this paper is one of the first papers that give probabilistic interpretation of Rogers (continuous $q-$ultraspherical) polynomials.