Characterizations of Shift-Invariant Distributions Based on Summation Modulo One

摘要

For n a member of N, let X,Y(1),...,Y(n) be independent random variables, andsuppose that X is distributed in ((bracket) 0,1), but not uniformly. The authors characterize the distributions of X and Y(s) (s=1,...,n) satisfying the equation (brace)X+Y(1)+...+Y(n)(brace) (equality in distribution) X, where (in braces: Z) denotes the fractional part of a random variable Z. In the case of full generality, Y(s) is lattice, and X is shift-invariant with respect to a discrete uniform distribution on ((bracket) 0,1). The authors also give a characterization of such shift invariant distributions. The authors consider some special cases of this equation: If X (equality in distribution) Y(1), then X has a shifted discrete uniform distribution on ((bracket) 0,1); further the case that Y(1),...,Y(n) are identically distributed, and a generalization of the equation with X,Y(1),...,Y(n) identically distributed is considered. These results generalize results of Goldman (1968) and of Arnold and Meeden (1976).