Orthogonality of the Sheffer system associated to a Lévy process (Q1973312)

The author studies multi-dimensional Lévy-Sheffer systems. He gives a construction of Sheffer polynomials associated to a \(d\)-dimensional Lévy process. The main result is a characterisation of orthogonal and pseudo-orthogonal Lévy-Sheffer systems (a family of polynomials \(P_n\) indexed by \({\mathbb N}^d\) is called pseudo-orthogonal if \(P_n\) and \(P_m\) are orthogonal whenever \(n_1+\cdots+n_d\not=m_1+\cdots+m_d\)). A Lévy-Sheffer system is pseudo-orthogonal if and only if the distributions of the Lévy process belong to the class of quadratic natural exponential families. Several examples are also given. For an exposition of the one-dimensional theory, see, e.g., Chapter 4 of the monograph [\textit{W. Schoutens}, ``Stochastic processes and orthogonal polynomials'' (2000; Zbl 0960.60076)].