Some Results on Distributions Arising From Coin Tossing

Abstract

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.