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Relative Entropy Densities and a Class of Limit Theorems of the Sequence of m-Valued Random Variables

Liu Wen
The Annals of Probability
Vol. 18, No. 2 (Apr., 1990), pp. 829-839
Stable URL: http://www.jstor.org/stable/2244318
Page Count: 11
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Relative Entropy Densities and a Class of Limit Theorems of the Sequence of m-Valued Random Variables
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Abstract

Let {Xn, n ≥ 1} be a sequence of random variables taking values in S = {1,2, ..., m} with distribution p(x1, ..., xn), (pi1, pi2, ..., pim), i = 1,2, ..., a sequence of probability distributions on S, and φn = (1/n)log p(X1, ..., Xn) - (1/n)∑n i = 1log piXi the entropy density deviation, relative to the distribution $\prod^n_{i = 1}p_{ix_i}, \text{of} \{X_i, 1 \leq i \leq n\}$. In this paper the relation between the limit property of φn and the frequency of given values in {Xn} is studied.

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