The Annals of Mathematical Statistics

On Discrete Variables whose Sum is Absolutely Continuous

David Blackwell

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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
links.jstor.org

Citation

Blackwell, David. On Discrete Variables whose Sum is Absolutely Continuous. Ann. Math. Statist. 28 (1957), no. 2, 520--521. doi:10.1214/aoms/1177706985. https://projecteuclid.org/euclid.aoms/1177706985


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