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The strong deviation theorem for discrete-time and continuous-state nonhomogeneous Markov chains
Journal of Inequalities and Applications volume 2013, Article number: 462 (2013)
Abstract
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.
1 Introduction
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 , .
Let
where ω is a sample point. In statistical terms, and are called the likelihood ratio and the asymptotic average log-likelihood ratio, respectively [1]. 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 [2]
Let be a nonhomogeneous Markov chain with the initial distribution , , and the transition probability density , , . If
this Markov chain is called a discrete-time and continuous-state nonhomogeneous Markov chain.
Let be a discrete-time and continuous-state nonhomogeneous Markov chain on the measure Q with the initial distribution density , and the transition probability density , , . Then for any , set B
then
so
There have been some works on deviation theorem, a kind of strong limit theorem represented by inequalities. Liu and Yang [3] have studied the limit properties of a class of averages of functions of two variables of arbitrary information sources. Liu and Yang [4] investigated the strong deviation theorems for arbitrary information source relative to Markov information source. Liu [5] 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 [6]. Liu and Wang [7] have studied a strong limit theorem expressed by inequalities for the sequences of absolutely continuous random variables. Recently, Fan [8] 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 [9], Liu and Yang [7] 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
Theorem 1 Let be a discrete-time and continuous-state nonhomogeneous Markov chain on the measure Q, and be defined by (1) and (2), respectively. Let be a sequence of set of the real line, and be the indicative function of . Let
and
Then
and
Proof Let λ be a nonnegative constant, and let
It is easy to see that is a density function of variables. Let
then is a nonnegative supermartingale that converges a.s. Hence there exists , such that
Letting in (9), we obtain
This implies that
We have by (7)
It follows from (2), (8), and (12) that
By (9) and (13), we have
(a) Let . Dividing the two sides of (14) by lnλ, we obtain
By (1) and (15), we have
By (3), (16), the property of the superior limit
and the inequality (), we have
By using the inequality (), we have by (17)
Let be the set of rational numbers in the interval , and let
Then we have by (18),
Let . It is easy to see if , as a function of λ attains its smallest value on the interval , and is increasing on the interval and . For each if , take , such that , we have
By (20), we have
By (21) and (22), we have
If , (23) holds obviously. Since , (4) holds by (23) when .
When , letting in (20), we have
Since , (4) also holds by (24) when .
(b) Let . Dividing the two sides of (14) by lnλ, we have
By (1) and (25), we have
By (26), (3), the property of the inferior limit
and the inequality (), we have
By using the inequality and (), we have by (27)
Let be the set of rational numbers in the interval , and let
Then we have by (28)
Let . It is easy to see that if , then as a function of λ attains its largest value on the interval , and is increasing on the interval and , and is decreasing on the interval and . For each , take , such that . Then we have
By (30), we have
By (31) and (32),
Since , (5) holds by (33) when .
When , for , hence we have by (30)
since , (5) also holds by (34) when .
It is easy to see that when , as a function of λ is decreasing on the interval and . For each , when , take , , such that . We have
By (30), we have
It follows from (35) and (36) that
when , (37) also holds obviously. Since , (6) follows from (37) directly. □
References
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Acknowledgements
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).
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BW carried out the study of strong deviation theorem and drafted the manuscript. ZS participated in the proof of theorem. All authors read and approved the final manuscript.
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Wang, B., Shi, Z. The strong deviation theorem for discrete-time and continuous-state nonhomogeneous Markov chains. J Inequal Appl 2013, 462 (2013). https://doi.org/10.1186/1029-242X-2013-462
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DOI: https://doi.org/10.1186/1029-242X-2013-462