Strong Decomposition of Random Variables |
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| Authors: | Jørgen Hoffmann-Jørgensen Abram M Kagan Loren D Pitt Lawrence A Shepp |
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| Institution: | (1) Department of Mathematical Sciences, University of Aarhus, 8000 Aarhus C, Denmark;(2) Department of Mathematics, University of Maryland, College Park, MD 20742, USA;(3) Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA;(4) Department of Statistics, Rutgers University, Piscataway, NJ 08855, USA |
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| 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. |
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| Keywords: | Absolute continuity Component Fisher information Singularity |
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