Orthogonality of the Sheffer system associated to a Lévy process
type: Item
instance of: scholarly article
title: Orthogonality of the Sheffer system associated to a Lévy process
zbMATH Open document ID: 0961.60024
DOI: 10.1016/S0378-3758(99)00158-5
published in: Journal of Statistical Planning and Inference
publication date: 2001-05-21T00:00:00Z
review text: 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)].
reviewed by: Uwe Franz
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 62E10
Mathematics Subject Classification ID: 62H05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1436953
zbMATH Keywords: Levy processes
zbMATH Keywords: Sheffer systems
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: exponential families
author: D. Pommeret
MaRDI profile type: Publication
cites work: Q4167890
cites work: Q4888858
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description: scientific article; zbMATH DE number 1436953