When Do Weighted Sums of Independent Random Variables have a Density--Some Results and Examples
| Autor: | Jakob I. Reich |
|---|---|
| Rok vydání: | 1982 |
| Předmět: |
Statistics and Probability
Exchangeable random variables Discrete mathematics distribution absolutely continuous Range splitting sequences of independent random variables E05 Random element weighted sums of range splitting sequences Convergence of random variables Sum of normally distributed random variables Statistics Probability and Uncertainty G50 Random variable G30 E10 singular with respect to Lebesgue measure Mathematics |
| Zdroj: | Ann. Probab. 10, no. 3 (1982), 787-798 |
| ISSN: | 0091-1798 |
| DOI: | 10.1214/aop/1176993787 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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