$C^\infty$ Densities for Weighted Sums of Independent Random Variables

Abstract

Let $\{X_n\}$ be a sequence of independent random variables and $\{a_n\}$ a square summable, positive nonincreasing sequence of real numbers such that $\sum a_n X_n$ is a random variable. We show that the condition $\lim_{n\rightarrow\infty} a^2_n \log(a_n)/\sum^\infty_{k=n+1} a^2_k = 0$ implies that the distribution measure $F(dx) = P(\sum a_n X_n \in dx)$ has an infinitely differentiable density for every range-splitting sequence $\{X_n\}$. The class of range-splitting sequences includes all non-trivial i.i.d. sequences with mean 0 and finite second moments. Consequences and examples are discussed.