Acta mathematica scientia,Series A ›› 2009, Vol. 29 ›› Issue (2): 517-527.

• Articles • Previous Articles    

A Class of Small Deviation Theorems for the Sequences of N-valued Random Variables with Respect to
mth-order Nonhomogeneous Markov Chains

  

  1. (Faculty of Science, Jiangsu University, Jiangsu Zhenjiang 212013)
  • Received:2007-08-03 Revised:2008-09-24 Online:2009-04-25 Published:2009-04-25
  • Supported by:

    国家自然科学基金(10571076)资助

Abstract:

Let {Xn, n ≥ 0} be a sequence of random variables on the probability space (Ω, F, P) taking values in alphabet S={1, 2, … , N}. Let Q be another probability measure on F, under which {Xn, n ≥ 0} is an mth-order onhomogeneous Markov chain. Let h(P|Q) be the sample divergence rate of P with respect to Q related to {Xn}. In this paper, the author first establishes a class of small deviation theorems for the averages of the functions of m+1 variables of {Xn}, n ≥ 0} with respect to $m$th-order nonhomogeneous Markov chains. As corollaries, the author obtains the small deviation theorems for the frequency of occurrence of the states and the entropy density of {Xn}, n ≥ 0} with respect to mth-order nonhomogeneous Markov chains. Finally, the author gets several strong laws of large numbers and a Shannon-McMillan theorem for mth-order nonhomogeneous Markov chains.

Key words: Small deviation theorems, Markov chains, Shannon-McMillan theorem

CLC Number: 

  • 60F15
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