论文信息 - Orthogonality of the Sheffer system associated to a Levy process

Orthogonality of the Sheffer system associated to a Levy process

Abstract The aim of this paper is to relate some recent results on Levy processes (see Schoutens and Teugels, 1998. Comm. Statist. Stochastic Models 14, 335–349) to a recent study of the author (1996) on multidimensional natural exponential families. In this way, we consider a natural construction of Sheffer polynomials associated to a d -dimensional Levy process and we prove that this is the only one that leads to an orthogonal Sheffer system. It is also shown that the orthogonality occurs if and only if the Levy process wanders throught the class of quadratic natural exponential families. Some interesting martingale properties are reviewed in a multidimensional setting.

Denys Pommeret | D. Pommeret

[1] C. Morris. Natural Exponential Families with Quadratic Variance Functions , 1982 .

[2] G. Letac. Le problem de la classification des familles exponentielles naturelles de ℝd ayant une fonction variance quadratique , 1989 .

[3] The $2d+4$ simple quadratic natural exponential families on ${\bf R}\sp d$ , 1996 .

[4] I. Sheffer. Concerning Appell sets and associated linear functional equations , 1937 .

[5] J. Meixner,et al. Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion , 1934 .

[6] Some classes of orthogonal polynomials associated with martingales , 1986 .

[7] Rene F. Swarttouw,et al. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue Report Fac , 1996, math/9602214.

[8] D. Pommeret. Orthogonal polynomials and natural exponential families , 1996 .

[9] O. Barndorff-Nielsen. Information And Exponential Families , 1970 .

[10] J. Teugels,et al. Lévy processes, polynomials and martingales , 1998 .