The Asymptotic Equipartition Property for a Nonhomogeneous Markov Information Source

Abstract

In this paper, we study the asymptotic equipartition property (AEP) for a nonhomogeneous Markov information source. We first give a limit theorem for the averages of the functions of two variables of this information source by using the convergence theorem for the martingale difference sequence. As corollaries, we get several limit theorems and a limit theorem of the relative entropy density, which hold for any nonhomogeneous Markov information source. Then, we get a class of strong laws of large numbers for nonhomogeneous Markov information sources. Finally, we prove the AEP for a class of nonhomogeneous Markov information sources.

A class of small deviation theorems for the random fields on an m rooted Cayley tree

“…Lemma 2 (see [16]). LetQ 1 be an m-dimensional stochastic matrix determined by the mth-order transition matrix Q 1 .…”

Some limit theorems for the second-order Markov chains indexed by a general infinite tree with uniform bounded degree

“…Lemma 2 (see [12]) LetP be a two-dimensional stochastic matrix determined by the second-order transition matrix P. If the elements of P are all positive, i.e.,…”

The Asymptotic Equipartition Property for Asymptotic Circular Markov Chains

“…Lemma 1 [8]: Let {X n , n ≥ 0} be a nonhomogeneous Markov chain with initial distribution (3) and transition matrixes (4); let f n (x, y)(n ≥ 1) be the real functions defined on S × S. If f n (x, y) is uniformly bounded, then…”

“…Yang [10] also studied another strong law of large numbers for bivariate functions of countable nonhomogeneous Markov chains under the condition of uniform convergence in the Cesàro sense of Markov chains. Yang [8] studied the asymptotic equipartition property for nonhomogeneous Markov chains. In practical applications, the case that the transition matrixes of nonhomogeneous Markov chains are asymptotic circular usually occurs, so studying the asymptotic circular Markov chains makes sense.…”