## The Annals of Mathematical Statistics

### Random Variables with Independent Binary Digits

George Marsaglia

#### 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

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

Digital Object Identifier
doi:10.1214/aoms/1177693058

Mathematical Reviews number (MathSciNet)
MR298715

Zentralblatt MATH identifier
0239.60015

JSTOR