June, 1957 On Discrete Variables whose Sum is Absolutely Continuous
David Blackwell
Ann. Math. Statist. 28(2): 520-521 (June, 1957). DOI: 10.1214/aoms/1177706985

## 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.

## Citation

David Blackwell. "On Discrete Variables whose Sum is Absolutely Continuous." Ann. Math. Statist. 28 (2) 520 - 521, June, 1957. https://doi.org/10.1214/aoms/1177706985

## Information

Published: June, 1957
First available in Project Euclid: 27 April 2007

zbMATH: 0078.31602
MathSciNet: MR88091
Digital Object Identifier: 10.1214/aoms/1177706985