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NECESSARY CONDITIONS FOR CHARACTERIZATION OF

LAWS VIA MIXED SUMS

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ANTHONY G. PAKES

*Department of Mathematics, University of Western Australia,*

Nedlands, WA, 6009, Australia
(Received July 5, 1993; revised May 12, 1994)

**Abstract.**
Suppose *X* and *Y* are independent and
identically distributed, and independent of *U* which satisfies 0
__<__ *U* __<__ 1. Recent work has centered on finding the laws \cal
L (*X*) for which *X* \cong *U*(*X*+*Y*) where \cong 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 Lévy 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.

*Key words and phrases*:
Distribution theory,
characterization, semi-stable
laws, mixtures, self-similar processes.

**Source**
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