Characterizations of shift-invariant distributions based on summation modulo one

R.J.G. Wilms, J.G.F. Thiemann

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    Abstract

    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.
    Original languageEnglish
    Place of PublicationEindhoven
    PublisherTechnische Universiteit Eindhoven
    Number of pages12
    Publication statusPublished - 1993

    Publication series

    NameMemorandum COSOR
    Volume9302
    ISSN (Print)0926-4493

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