Multivariate Meixner classes of invariant distributions (Q1819497)

The univariate Meixner classes of distributions were first introduced by \textit{J. Meixner} [cf. J. Lond. Math. Soc. 9, 6-13 (1934; Zbl 0008.16205)] and their properties and characterizations studied in the literature. In this paper multivariate Meixner classes of invariant distributions of random matrices X are introduced and their properties studied. Four classes are defined according as X is continuous (or discrete) and symmetric (or non-symmetric). To save space we provide the definitions and some results for the case when X is continuous and symmetric. Let X be a centered, symmetric, continuous \(p\times p\) random matrix with a moment generating function (mgf) \(\psi\) and a system of orthogonal polynomials (o.p.) \(\{P_ k\}\) where each \(P_ k\) is indexed by a partition \(\sigma\) of an integer k, \(k=0,1,... \). X is said to be invariant if the distribution of X is invariant under the transformation from X to HXH', H in \(O_ p\). Here \(O_ p\) is the orthogonal group of \(p\times p\) orthonormal matrices. X is said to belong to a continuous symmetric Meixner class defined by U(T), T a \(p\times p\) symmetric matrix, if the gf for the associated o.p.'s is of the form \[ K(X;T)=\sum^{\infty}_{k=0}1/k!\sum_{\sigma}P_{\sigma}(X)C_{\sigma}(T)/C_{\sigma}(I)=\int_{O_ p}\exp (tr[HXH'U(T)])dH/\psi (U(T)). \] Here \(C_{\sigma}(T)\) are the zonal polynomials defined for the ordered partitions \(\sigma\) of k into not more than p parts. See \textit{A. T. James} [Ann. Math. Stat. 35, 475-501 (1964; Zbl 0121.366)] and \textit{A. G. Constantine} [ibid. 34, 1270-1285 (1963; Zbl 0123.368)] for a detailed discussion of these polynomials. A sample of the results: (1) The centered Wishart distribution is a continuous symmetric Meixner class defined by \(U(T)=-T(I-T)^{-1}\). (2) (a) If \(X_ 1\) and \(X_ 2\) are independent and belong to the same multivariate Meixner class then so does \(X=X_ 1+X_ 2\) and (*) \(K(X;T)=K(X_ 1;T)K(X_ 2;T)\). (This is a multivariate generalization of Runge's identity.) (b) Conversely if \(X_ 1\) and \(X_ 2\) are independent and invariant under the orthogonal transformation and possess mgf's and (*) holds then \(X_ 1\) and \(X_ 2\) belong to the same multivariate Meixner class.