The Annals of Probability

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

Jakob I. Reich

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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.

Article information

Source
Ann. Probab., Volume 10, Number 3 (1982), 787-798.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993787

Digital Object Identifier
doi:10.1214/aop/1176993787

Mathematical Reviews number (MathSciNet)
MR659548

Zentralblatt MATH identifier
0484.60012

JSTOR
links.jstor.org

Keywords
E05 G30 G50 E10 Range splitting sequences of independent random variables weighted sums of range splitting sequences distribution absolutely continuous singular with respect to Lebesgue measure

Citation

Reich, Jakob I. When Do Weighted Sums of Independent Random Variables have a Density--Some Results and Examples. Ann. Probab. 10 (1982), no. 3, 787--798. doi:10.1214/aop/1176993787. https://projecteuclid.org/euclid.aop/1176993787


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