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Two-dimensional Meixner Random Vectors of Class ${\mathcal{M}}_{L}$

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  • Aurel I. Stan

    (The Ohio State University at Marion)

Abstract

The paper is divided into two parts. In the first part we lay down the foundation for defining the joint annihilation–preservation–creation decomposition of a finite family of not necessarily commutative random variables, and show that this decomposition is essentially unique. In the second part we show that any two, not necessarily commutative, random variables X and Y for which the vector space spanned by the identity and their annihilation, preservation, and creation operators equipped with the bracket given by the commutator forms a Lie algebra are equivalent up to an invertible linear transformation to two independent Meixner random variables with mixed preservation operators. In particular, if X and Y commute, then they are equivalent up to an invertible linear transformation to two independent classic Meixner random variables. To show this we start with a small technical condition called “non-degeneracy”.

Suggested Citation

  • Aurel I. Stan, 2011. "Two-dimensional Meixner Random Vectors of Class ${\mathcal{M}}_{L}$," Journal of Theoretical Probability, Springer, vol. 24(1), pages 39-65, March.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:1:d:10.1007_s10959-010-0309-4
    DOI: 10.1007/s10959-010-0309-4
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    References listed on IDEAS

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    1. D. Pommeret, 1996. "Orthogonal polynomials and natural exponential families," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 5(1), pages 77-111, June.
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