On the Modular Value and Fractional Part of a Random Variable

Abstract

Let X be a random variable with characteristic function ϕ. In the case where X is integer-valued and n is a positive integer, a formula (in terms of ϕ) for the probability that n divides X is presented. The derivation of this formula is quite simple and uses only the basic properties of expectation and complex numbers. The formula easily generalizes to one for the distribution of X mod n. Computational simplifications and the relation to the inversion formula are also discussed; the latter topic includes a new inversion formula when the range of X is finite.When X may take on a more general distribution, limiting considerations of the previous formulas suggest others for the distribution, density, and moments of the fractional part X — [X]. These are easily derived using basic properties of Fourier series. These formulas also yield an alternative inversion formula for ϕ when the range of X is bounded.Applications to renewal theory and random walks are suggested. A by-product of the approach is a probabilistic method for the evaluation of infinite series.