The Annals of Probability

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

Liu Wen

Abstract

Let $\{X_n, n \geq 1\}$ be a sequence of random variables taking values in $S = \{1,2, \ldots, m\}$ with distribution $p(x_1, \ldots, x_n), (p_{i1}, p_{i2}, \ldots, p_{im}), i = 1,2, \ldots$, a sequence of probability distributions on $S$, and $\varphi_n = (1/n)\log p(X_1, \ldots, X_n) - (1/n)\sum^n_{i = 1}\log p_{iX_i}$ 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 $\varphi_n$ and the frequency of given values in $\{X_n\}$ is studied.

Article information

Source
Ann. Probab., Volume 18, Number 2 (1990), 829-839.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176990860

Digital Object Identifier
doi:10.1214/aop/1176990860

Mathematical Reviews number (MathSciNet)
MR1055435

Zentralblatt MATH identifier
0711.60026

JSTOR

Subjects
Primary: 60F15: Strong theorems
Secondary: 94A17: Measures of information, entropy

Citation

Wen, Liu. Relative Entropy Densities and a Class of Limit Theorems of the Sequence of $m$-Valued Random Variables. Ann. Probab. 18 (1990), no. 2, 829--839. doi:10.1214/aop/1176990860. https://projecteuclid.org/euclid.aop/1176990860