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We study properties of compactly supported, 4 parameter \newline $(\rho_{12},\rho _{23},\rho _{13},q)\in (-1,1)^{\times 4}$ family of continuous type3 dimensional distributions, that have the property that for $q\rightarrow1^{-}$ this family tends to some 3 dimensional Normal distribution. For $q=0$we deal with 3 dimensional generalization of Kesten--McKay distribution. In avery special case when $\rho _{12}\rho _{13}\rho _{23}=q$ all one dimensionalmarginals are identical, semicircle distributions. We find both all marginal aswell as all conditional distributions. Moreover, we find also families ofpolynomials that are orthogonalized by these one-dimensional margins andone-dimensional conditional distributions. Consequently, we find moments ofboth conditional and unconditional distributions of dimensions one and two. Inparticular, we show that all one-dimensional and two-dimensional conditionalmoments of, say, order $n$ and are polynomials of the same order $n$ in theconditioning random variables. Finding above mentioned orthogonal polynomialsleads us to a probabilistic interpretation of these polynomials. Among them arethe famous Askey-Wilson, Al-Salam--Chihara polynomials considered in thecomplex, but conjugate, parameters, as well as q-Hermite and Rogerspolynomials. It seems that this paper is one of the first papers that give aprobabilistic interpretation of Rogers (continuous q-ultraspherical)polynomials.

journal of mathematical analysis and applications

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