Strong Decomposition of Random Variables

Jørgen Hoffmann-Jørgensen (), Abram M. Kagan (), Loren D. Pitt () and Lawrence A. Shepp ()
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Jørgen Hoffmann-Jørgensen: University of Aarhus
Abram M. Kagan: University of Maryland
Loren D. Pitt: University of Virginia
Lawrence A. Shepp: Rutgers University

Journal of Theoretical Probability, 2007, vol. 20, issue 2, 211-220

Abstract: 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.

Keywords: Absolute continuity; Component; Fisher information; Singularity (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/s10959-007-0061-6

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