The Annals of Mathematical Statistics

On Discrete Variables whose Sum is Absolutely Continuous

David Blackwell

Abstract

If $\{Z_n\}, n = 1, 2, \ldots$ is a stationary stochastic process with $D$ states $0, 1, \cdots, D - 1$, and $X = \sum^\infty_1 Z_k/D^n$, Harris [1] has shown that the distribution of $X$ is absolutely continuous if and only if the $Z_n$ are independent and uniformly distributed over $0, 1, \cdots, D - 1$, i.e., if and only if the distribution of $X$ is uniform on the unit interval. In this note we show that if $\{Z_n\}, n = 1, 2, \cdots$ is any stochastic process with $D$ states $0, 1, \cdots, D - 1$ such that $X = \sum^\infty_1 Z_n/D^n$ has an absolutely continuous distribution, then the conditional distribution of $R_k = \sum^\infty_{n = 1} Z_{k + n}/D^n$ given $Z_1, \cdots, Z_k$ converges to the uniform distribution on the unit interval with probability 1 as $k \rightarrow \infty$. It follows that the unconditional distribution of $R_k$ converges to the uniform distribution as $k \rightarrow \infty$. Since if $\{Z_n\}$ is stationary the distribution of $R_k$ is independent of $k$, the result of Harris follows.

Article information

Source
Ann. Math. Statist., Volume 28, Number 2 (1957), 520-521.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177706985

Digital Object Identifier
doi:10.1214/aoms/1177706985

Mathematical Reviews number (MathSciNet)
MR88091

Zentralblatt MATH identifier
0078.31602

JSTOR