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Section  Mathematics 
Title  On the limit distribution of a number of runs in polynomial sequence controlled by Markov chain 
Author(s)  Mezhennaya N.M.^{a} 
Affiliations  Bauman Moscow State Technical University^{a} 
Abstract  The present paper is devoted to studying the asymptotic properties of a number of runs in the sequence of discrete random variables controlled by Markov chain with a finite number of states. A chain state at each step determines the law of characters distribution in the controlled sequence at this step. This random sequence represents a model of hidden Markov chain. Using ChenStein method we estimate the total variation distance between the distribution of the number of runs with length not less than predetermined length in the random sequence controlled by Markov chain and the accompanying Poisson distribution. For this purpose we first consider the sequence of independent inhomogeneous polynomial random variables, and then we use an approach which allows to get the estimate for total variation distance between mixed Poisson distribution and Poisson distribution with the parameter which equals to an average number of runs with length not less than predetermined. The estimate is based on both the variance of the mixed Poisson distribution parameter and the estimate obtained earlier for the total variation distance for the polynomial scheme. Separately we consider the case of a stationary Markov chain. Using derived estimates we investigate Poisson and normal limit theorems for the number of runs with length not less than predetermined, as well as the limit distribution for the maximal run length in a controlled sequence. 
Keywords  Markov chain, polynomial random sequence, number of runs, Poisson limit theorem, total variation distance, ChenStein method 
UDC  519.214.5, 519.217.2 
MSC  60F05, 60B10, 60J10 
DOI  10.20537/vm160303 
Received  23 May 2016 
Language  Russian 
Citation  Mezhennaya N.M. On the limit distribution of a number of runs in polynomial sequence controlled by Markov chain, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 3, pp. 324335. 
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