NECESSARY CONDITIONS FOR CHARACTERIZATION OF LAWS VIA MIXED SUMS

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 L(X) for which X ≅ U(X + Y) where ≅ 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.