Let $\{X_n\}$ be a sequence of independent identically distributed random variables which take the values $\pm 1$ with probability $\frac{1}{2}$. Let $X = \sum^\infty_{n=1} a_nX_n$ where $\sum a^2_n < \infty$. We show that if $n^{_\alpha} \leq |a_n| \leq n^{-\beta}$ for some $\alpha > \frac{1}{2}$ and $0 \leq \alpha - \beta < \frac{1}{2}$ then the distribution of $X = \sum a_nX_n$ is absolutely continuous with respect to Lebesgue measure. We then prove similar results for more general independent sequences. We also show that if $\lim\inf 2^N \sqrt{\sum^\infty_n=N+1} a^2_n = 0$ then the distribution of $X = \sum a_nX_n$ is singular with respect to Lebesgue measure.
Publié le : 1982-08-14
Classification:
Sums of independent random variables,
distributions absolutely continuous or singular with respect to Lebesgue measure,
60E05
@article{1176993786,
author = {Reich, Jakob I.},
title = {Some Results on Distributions Arising From Coin Tossing},
journal = {Ann. Probab.},
volume = {10},
number = {4},
year = {1982},
pages = { 780-786},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993786}
}
Reich, Jakob I. Some Results on Distributions Arising From Coin Tossing. Ann. Probab., Tome 10 (1982) no. 4, pp. 780-786. http://gdmltest.u-ga.fr/item/1176993786/