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Orthogonality of the Sheffer system associated to a Lévy process. (English) Zbl 0961.60024
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 [W. Schoutens, “Stochastic processes and orthogonal polynomials” (2000; Zbl 0960.60076)].

##### MSC:
 60E07 Infinitely divisible distributions; stable distributions 62E10 Characterization and structure theory of statistical distributions 62H05 Characterization and structure theory for multivariate probability distributions; copulas 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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##### References:
 [1] Barndorff-Nielsen, O.E., 1978. Information and Exponential Families. Wiley, New York. · Zbl 0387.62011 [2] Bertoin, J., 1996. Lévy Processes. Cambridge University Press, Cambridge. [3] Casalis, M., The 2d+4 simple quadratic natural exponential families on $$R\^{}\{d\}$$, Ann. statist., 24, 1828-1854, (1996) · Zbl 0867.62042 [4] Feinsilver, P., Some classes of orthogonal polynomials associated with martingales, Proc. AMS, 98, 298-302, (1986) · Zbl 0615.60050 [5] Koekoek, R., Swarttouw, R.F., 1994. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Report no. 94-05. Delft University of Technology, Faculty of Technical Mathematics and Informatics. [6] Letac, G., 1989. Le problème de la classification des familles exponentielles naturelles de $$R\^{}\{d\}$$ ayant une fonction variance quadratique. In: Probability Measure on Groups IX. Lecture Notes in Mathematics, Vol. 1306. Springer, Berlin, pp. 194-215. [7] Letac, G., 1992. Lectures on Natural Exponential Families and their Variance Functions. Instituto de matemática pura e aplicada: Monografias de matemática, Vol. 50, Rı́o de Janeiro, Brésil. [8] Meixner, J., Orthogonal polynomsysteme mit einer besonderen gestalt der erzengenden function, J. London math. soc., 9, 6-13, (1934) · JFM 60.0293.01 [9] Morris, C.N., Natural exponential families with quadratic variance functions, Ann. statist., 10, 65-80, (1982) · Zbl 0498.62015 [10] Pommeret, D., Orthogonal polynomials and natural exponential families, Test, 5, 77-111, (1996) · Zbl 0960.62517 [11] Sheffer, I.M., Concerning Appell sets and associated linear functional equations, Duke math. J., 3, 593-609, (1937) · Zbl 0018.13601 [12] Schoutens, W.; Teugels, J.L., Lévy processes, polynomials and martingales, Comm. statist. stochastic models, 14, 335-349, (1998) · Zbl 0895.60050
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