When Do Weighted Sums of Independent Random Variables have a Density--Some Results and Examples

Abstract

Let $\{X_n\}$ be a sequence of independent random variables and $\{a_n\}$ a positive decreasing sequence such that $\sum a_nX_n$ is a random variable. We show that under mild conditions on $\{X_n\}$ (i) if for every $\delta, \lambda > 0$ $\sum^\infty_{n=1} \int^{\delta/a_{n+1}}_{\delta/a_n} \exp(-\lambda\xi^2 \sum^\infty_{k=n+1} a^2_k) d\xi < \infty$ then $P(\sum a_nX_n \in dx)$ has a density. (ii) $\lim_{\xi \rightarrow \infty}|E(e^{i\xi\sum a_nX_n})| = 0$ for every $\{X_n\} \operatorname{iff} \lim_{N\rightarrow\infty} a^{-2}_N \sum^\infty_{n=N + 1} a^2_n = \infty.$ Several consequences, examples and counterexamples are given.