## The Annals of Probability

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

Jakob I. Reich

#### 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.

#### Article information

Source
Ann. Probab., Volume 14, Number 3 (1986), 1005-1013.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176992454

Digital Object Identifier
doi:10.1214/aop/1176992454

Mathematical Reviews number (MathSciNet)
MR841600

Zentralblatt MATH identifier
0593.60023

JSTOR
Reich, Jakob I. $C^\infty$ Densities for Weighted Sums of Independent Random Variables. Ann. Probab. 14 (1986), no. 3, 1005--1013. doi:10.1214/aop/1176992454. https://projecteuclid.org/euclid.aop/1176992454