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Strong Decomposition of Random Variables

Author

Listed:
  • Jørgen Hoffmann-Jørgensen

    (University of Aarhus)

  • Abram M. Kagan

    (University of Maryland)

  • Loren D. Pitt

    (University of Virginia)

  • Lawrence A. Shepp

    (Rutgers University)

Abstract

A random variable X is called strongly decomposable into (strong) components Y,Z, if X=Y+Z where Y=φ(X), Z=X−φ(X) are independent nondegenerate random variables and φ is a Borel function. Examples of decomposable and indecomposable random variables are given. It is proved that at least one of the strong components Y and Z of any random variable X is singular. A necessary and sufficient condition is given for a discrete random variable X to be strongly decomposable. Phenomena arising when φ is not Borel are discussed. The Fisher information (on a location parameter) in a strongly decomposable X is necessarily infinite.

Suggested Citation

  • Jørgen Hoffmann-Jørgensen & Abram M. Kagan & Loren D. Pitt & Lawrence A. Shepp, 2007. "Strong Decomposition of Random Variables," Journal of Theoretical Probability, Springer, vol. 20(2), pages 211-220, June.
    • Handle: RePEc:spr:jotpro:v:20:y:2007:i:2:d:10.1007_s10959-007-0061-6
    DOI: 10.1007/s10959-007-0061-6
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    References listed on IDEAS

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    1. Ostrovskii, I. V., 1977. "The arithmetic of probability distributions," Journal of Multivariate Analysis, Elsevier, vol. 7(4), pages 475-490, December.
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