The strong deviation theorem for discrete-time and continuous-state nonhomogeneous Markov chains
© Wang and Shi; licensee Springer. 2013
Received: 22 May 2013
Accepted: 9 September 2013
Published: 7 November 2013
In this paper, the notion of asymptotic average log-likelihood ratio, as a measure of the difference between the sequence of random variables and Markov chains, is introduced, and by constructing a nonnegative martingale, the strong deviation theorem for discrete-time and continuous-state nonhomogeneous Markov chains is established.
Let be the probability space, and let be a sequence of continuous random variables taking values in R and with the joint density function , . Let Q be another probability measure on , and be an independent random sequence on the measure Q, with the joint density function , .
where ω is a sample point. In statistical terms, and are called the likelihood ratio and the asymptotic average log-likelihood ratio, respectively . Obviously, if , , then , a.s. So can be used as a measure of deviation between and when n tends to infinity. The smaller is, the smaller the deviation is.
Definition 1 
this Markov chain is called a discrete-time and continuous-state nonhomogeneous Markov chain.
There have been some works on deviation theorem, a kind of strong limit theorem represented by inequalities. Liu and Yang  have studied the limit properties of a class of averages of functions of two variables of arbitrary information sources. Liu and Yang  investigated the strong deviation theorems for arbitrary information source relative to Markov information source. Liu  discussed a class of strong deviation theorems for an arbitrary stochastic sequence with respect to the marginal distribution by using generating function method, and also studied the problem above by means of Laplace transform . Liu and Wang  have studied a strong limit theorem expressed by inequalities for the sequences of absolutely continuous random variables. Recently, Fan  has studied some strong deviation theorems for dependent continuous random sequence.
In this paper, by using the notion of asymptotic log-likehood and the martingale convergence theorem, and extending the analytic technique proposed by Liu , Liu and Yang  to the case of discrete-time and continuous-state nonhomogeneous Markov chains, we obtain the strong deviation theorem for discrete-time and continuous-state nonhomogeneous Markov chains.
2 Main result
If , (23) holds obviously. Since , (4) holds by (23) when .
Since , (4) also holds by (24) when .
Since , (5) holds by (33) when .
since , (5) also holds by (34) when .
when , (37) also holds obviously. Since , (6) follows from (37) directly. □
This work is supported by the National Natural Science Foundations of China (11071104, 11226210), and the Research Foundation for Advanced Talents of Jiangsu University (11JDG116).
- Liu W: Strong Deviation and Analytic Method. Science Press, Beijing; 2003.Google Scholar
- Wang B: Strong law of large numbers for discrete-time and continuous-state nonhomogeneous Markov chains. J. Jiangsu Univ. Nat. Sci. 2008, 29(1):86–88.MATHGoogle Scholar
- Liu W, Yang WG: The limit properties of a class of averages of functions of two variables of arbitrary information sources. Chinese J. Appl. Probab. Statist. 1995, 11(2):195–203.MathSciNetMATHGoogle Scholar
- Liu W, Yang WG: A comparison between arbitrary information sources and nonhomogeneous Markov information sources and some small deviations theorems. Acta Math. Sin. (Chin. Ser.) 1997, 40(1):22–36.MATHGoogle Scholar
- Liu W, Yang WG: The Markov approximation of the sequences of N -valued random variables and a class of small deviation theorems. Stoch. Process. Appl. 2000, 89: 117–130. 10.1016/S0304-4149(00)00016-8View ArticleMathSciNetMATHGoogle Scholar
- Liu W: A class of strong deviation theorems and Laplace transform methods. Chin. Sci. Bull. 1998, 43(10):1036–1041.MathSciNetGoogle Scholar
- Liu W, Wang YJ: A strong limit theorem expressed by inequalities for the sequences of absolutely continuous random variables. Hiroshima Math. J. 2002, 32: 379–387.MathSciNetMATHGoogle Scholar
- Fan AH: Some strong deviation theorems for dependent continuous random sequence. Chin. Q. J. Math. 2010, 25(4):572–577.MATHGoogle Scholar
- Liu W: Relative entropy densities and a class of limit theorems of the sequences of m -valued random variables. Ann. Probab. 1990, 18: 829–839. 10.1214/aop/1176990860MathSciNetView ArticleMATHGoogle Scholar
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