## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 42, Number 6 (1971), 1922-1929.

### Random Variables with Independent Binary Digits

#### Abstract

Let $X = \cdot b_1b_2b_3 \cdots$ be a random variable with independent binary digits $b_n$ taking values 0 or 1 with probability $p_n$ and $q_n = 1 - p_n$. When does $X$ have a density? A continuous density? A singular distribution? This note gives necessary and sufficient conditions for the distribution of $X$ to be: discrete: $\Sigma\min (p_n, q_n) < \infty$; singular: $\Sigma^\infty_m\lbrack\log (p_n/q_n)\rbrack^2 = \infty$ for every $m$; absolutely continuous: $\Sigma^\infty_m\lbrack\log (p_n/q_n)\rbrack^2 < \infty$ for some $m$. Furthermore, $X$ has a density that is bounded away from zero on some interval if and only if $\log (p_n/q_n)$ is a geometric sequence with ratio $\frac{1}{2}$ for $n > k$, and in that case the fractional part of $2^k X$ has an exponential density (increasing or decreasing with the uniform a special case).

#### Article information

**Source**

Ann. Math. Statist. Volume 42, Number 6 (1971), 1922-1929.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177693058

**Digital Object Identifier**

doi:10.1214/aoms/1177693058

**Mathematical Reviews number (MathSciNet)**

MR298715

**Zentralblatt MATH identifier**

0239.60015

**JSTOR**

links.jstor.org

#### Citation

Marsaglia, George. Random Variables with Independent Binary Digits. Ann. Math. Statist. 42 (1971), no. 6, 1922--1929. doi:10.1214/aoms/1177693058. https://projecteuclid.org/euclid.aoms/1177693058

#### See also

- Acknowledgment of Prior Result: George Marsaglia. Acknowledgment of Priority to "Random Variables with Independent Binary Digits". Ann. Probab., Volume 2, Number 4 (1974), 747.Project Euclid: euclid.aos/1176342776