The Annals of Probability

Entropy and the Consistent Estimation of Joint Distributions

Katalin Marton and Paul C. Shields

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The $k$th-order joint distribution for an ergodic finite-alphabet process can be estimated from a sample path of length $n$ by sliding a window of length $k$ along the sample path and counting frequencies of $k$-blocks. In this paper the problem of consistent estimation when $k = k(n)$ grows as a function of $n$ is addressed. It is shown that the variational distance between the true $k(n)$-block distribution and the empirical $k(n)$-block distribution goes to 0 almost surely for the class of weak Bernoulli processes, provided $k(n) \leq (\log n)/(H + \epsilon)$, where $H$ is the entropy of the process. The weak Bernoulli class includes the i.i.d. processes, the aperiodic Markov chains and functions thereof and the aperiodic renewal processes. A similar result is also shown to hold for functions of irreducible Markov chains. This work sharpens prior results obtained for more general classes of processes by Ornstein and Weiss and by Ornstein and Shields, which used the $\bar{d}$-distance rather than the variational distance.

Article information

Ann. Probab., Volume 22, Number 2 (1994), 960-977.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 28D20: Entropy and other invariants
Secondary: 60J05: Discrete-time Markov processes on general state spaces 62B20 60G10: Stationary processes 94A17: Measures of information, entropy

Empirical distribution entropy weak Bernoulli processes


Marton, Katalin; Shields, Paul C. Entropy and the Consistent Estimation of Joint Distributions. Ann. Probab. 22 (1994), no. 2, 960--977. doi:10.1214/aop/1176988736.

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