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Transorthogonal polynomials and simple cubic multivariate distributions. (English) Zbl 1130.62052

Summary: Transorthogonality for a sequence of polynomials on \(\mathbb R^d\) has been recently introduced in order to characterize the reference probability measures, which are multivariate distributions of the natural exponential families (NEFs) having a simple cubic variance function. The present paper pursues this characterization of three various manners through exponential generating functions, transdiagonality of Bhattacharyya matrices and semigroup-Sheffer systems, respectively. The obtained results extend those well-known of simple quadratic NEFs based on the classical orthogonality of associated polynomials. The transorthogonality property is then compared to the 2-pseudo-orthogonality one which globaly characterizes the cubic NEFs. Finally, some techniques of calculation of these polynomials are presented and then illustrated on a multivariate normal inverse Gaussian Lévy process.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
60G50 Sums of independent random variables; random walks
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
62E10 Characterization and structure theory of statistical distributions
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