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.
Publié le : 1986-07-14
Classification:
E05,
G30,
G50,
E10,
Range-splitting sequences of independent random variables,
weighted sums of range-splitting sequences,
infinitely differentiable densities
@article{1176992454,
author = {Reich, Jakob I.},
title = {$C^\infty$ Densities for Weighted Sums of Independent Random Variables},
journal = {Ann. Probab.},
volume = {14},
number = {4},
year = {1986},
pages = { 1005-1013},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992454}
}
Reich, Jakob I. $C^\infty$ Densities for Weighted Sums of Independent Random Variables. Ann. Probab., Tome 14 (1986) no. 4, pp. 1005-1013. http://gdmltest.u-ga.fr/item/1176992454/