Sum of uniformly distributed random variables and a family of nonalyticC ∞-functions

Abstract

It is shown that the probability density functionp n(x) for the random variableX n = α[U n + (1 − α)U n − 1 + ... + (1 − α) n − 2 U 2 + (1 − α) n − 1 U 1] (where 0<α<1 andU i ′ s are independent random variables subject to the uniform distribution on the interval [−1, 1]) uniformly converges to an infinitely differentiable functionp ∞(x) asn→∞, and thatp ∞(x) is nonanalytic at infinitely many points. In particular, for α=1/2,p ∞(x) is nowhere analytic on [−1, 1].