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July, 1986 $C^\infty$ Densities for Weighted Sums of Independent Random Variables
Jakob I. Reich
Ann. Probab. 14(3): 1005-1013 (July, 1986). DOI: 10.1214/aop/1176992454


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.


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Jakob I. Reich. "$C^\infty$ Densities for Weighted Sums of Independent Random Variables." Ann. Probab. 14 (3) 1005 - 1013, July, 1986.


Published: July, 1986
First available in Project Euclid: 19 April 2007

MathSciNet: MR841600
zbMATH: 0593.60023
Digital Object Identifier: 10.1214/aop/1176992454

Keywords: E05 , E10 , G30 , G50 , infinitely differentiable densities , Range-splitting sequences of independent random variables , weighted sums of range-splitting sequences

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 3 • July, 1986
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