Characterizations of some polynomial variance functions byd-pseudo-orthogonality

Abstract

From a notion ofd-pseudo-orthogonality for a sequence of polynomials (d ∈ {2,3, …}), this paper introduces three different characterizations of natural exponential families (NEF's) with polynomial variance functions of exact degree 2d-1. These results provide extended versions of the Meixner (1934), Shanbhag (1972), 1979) and Feinsilver (1986) characterization results of quadratic NEF's based on classical orthogonal polynomials. Some news sets of polynomials with (2d-1)-term recurrence relation are then pointed out and we completely illustrate the cases associated to the families of positive stable distributions.