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Distributional Transformations, Orthogonal Polynomials, and Stein Characterizations

Author

Listed:
  • Larry Goldstein

    (University of Southern California)

  • Gesine Reinert

    (University of Oxford)

Abstract

A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function’s higher order derivatives. The class includes the size and zero bias transformations, and when specializing to weighting by polynomial functions, relates distributional families closed under independent addition, and in particular the infinitely divisible distributions, to the family of transformations induced by their associated orthogonal polynomial systems. For these families, generalizing a well known property of size biasing, sums of independent variables are transformed by replacing summands chosen according to a multivariate distribution on its index set by independent variables whose distributions are transformed by members of that same family. A variety of the transformations associated with the classical orthogonal polynomial systems have as fixed points the original distribution, or a member of the same family with different parameter.

Suggested Citation

  • Larry Goldstein & Gesine Reinert, 2005. "Distributional Transformations, Orthogonal Polynomials, and Stein Characterizations," Journal of Theoretical Probability, Springer, vol. 18(1), pages 237-260, January.
    • Handle: RePEc:spr:jotpro:v:18:y:2005:i:1:d:10.1007_s10959-004-2602-6
    DOI: 10.1007/s10959-004-2602-6
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    Cited by:

    1. Marie Ernst & Yvik Swan, 2022. "Distances Between Distributions Via Stein’s Method," Journal of Theoretical Probability, Springer, vol. 35(2), pages 949-987, June.
    2. Christian Döbler, 2017. "Distributional Transformations Without Orthogonality Relations," Journal of Theoretical Probability, Springer, vol. 30(1), pages 85-116, March.

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