The Annals of Probability

Some Results on Distributions Arising From Coin Tossing

Jakob I. Reich

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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.

Article information

Source
Ann. Probab., Volume 10, Number 3 (1982), 780-786.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993786

Digital Object Identifier
doi:10.1214/aop/1176993786

Mathematical Reviews number (MathSciNet)
MR659547

Zentralblatt MATH identifier
0484.60011

JSTOR
links.jstor.org

Subjects
Primary: 60E05: Distributions: general theory

Keywords
Sums of independent random variables distributions absolutely continuous or singular with respect to Lebesgue measure

Citation

Reich, Jakob I. Some Results on Distributions Arising From Coin Tossing. Ann. Probab. 10 (1982), no. 3, 780--786. doi:10.1214/aop/1176993786. https://projecteuclid.org/euclid.aop/1176993786


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