ENTROPY AND THE CONSISTENT ESTIMATION OF JOINT DISTRIBUTIONS
成果类型：
Article
署名作者：
MARTON, K; SHIELDS, PC
署名单位：
University System of Ohio; University of Toledo; Eotvos Lorand University
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/aop/1176988736
发表日期：
1994
页码：
960-977
关键词：
摘要：
The kth-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) less-than-or-equal-to (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 d-distance rather than the variational distance.