Necessary conditions for characterization of laws via mixed sums

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Abstract

Suppose X and Y are independent and identically distributed, and independent of U which satisfies 0 less than or equal to U less than or equal to 1. Recent work has centered on finding the laws L(X) for which X congruent to U(X + Y) where congruent to denotes equality in law. We show that this equation corresponds to a certain projective invariance property under random rotations. Implicitly or explicitly, it has been assumed that the characteristic function of X has an expansion property near the origin. We show that solutions may be admitted in the absence of this condition when -log U has a lattice law. A continuous version of the basic problem replaces sums with a Levy process. Instead we consider self-similar processes, showing that a solution exists only when U is constant, and then all processes of a given order are admitted.
Original languageEnglish
Pages (from-to)797-802
JournalAnnals of the Institute of Statistical Mathematics
Volume46
DOIs
Publication statusPublished - 1994

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