Characterizations of shift-invariant distributions based on summation modulo one

For n \in N, let X, Y_1, ..., Y_n be independent random variables, and suppose that X is distributed in [0,1), but not uniformly. We characterize the distributions of X and Y_s (s=1,...,n) satisfying the equation $\{ X+Y_1+...+Y_n\} \stackrel{\rm{d}}{=} X$, where {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 unifonn distribution on [0,1). We also give a characterization of such shift-invariant distributions. In addition, we consider some special cases of this equation: If $X \stackrel{\rm{d}}{=} Y_1$, then X has a shifted discrete uniform distribution on [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. Our results generalize results of Goldman (1968) and of Arnold and Meeden (1976). Key words and phrases: Fourier-Stieltjes coefficients; distribution modulo 1; fractional parts.