Estimator for the distribution of the numbers of runs in a random sequence controlled by stationary Markov chain | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2017. № 35. DOI: 10.17223/20710410/35/2

The sequences of random characters from a finite set A with polynomial distributions controlled by a stationary finite-state Markov chain are considered. For numbers of character runs in them, the asymptotic properties of joint distributions are studied. We deduce an estimate for the total variation distance p_TV between the distribution of a random vector with components being numbers of runs in a controlled sequence of an enough length T and accompanying multidimensional Poisson distribution Pois(A_A). The estimate is p_TV (L(A)) ^ y (7T(p*)* +1), where Y = |A|(2s* + 3)(p*)*, s* (s*) is the minimum (maximum) length of run in the set of components of the vector ^д, and p* is the maximum character probability in distributions given on A. For deriving this estimate, we use the functional variant of Chen - Stein method and an estimation for the total variation distance between the mixed and ordinal Poisson distributions. This estimation is a function of the variance of mixing parameter of mixed Poisson distribution. Using the derived estimate for the total variation distance p_TV, we deduce the multidimensional Poisson and normal limit theorems for the random vector under appropriate conditions for scheme parameters.

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Title Estimator for the distribution of the numbers of runs in a random sequence controlled by stationary Markov chain
Headline Estimator for the distribution of the numbers of runs in a random sequence controlled by stationary Markov chain
Publesher Tomsk State University
Issue Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics 35 3/2017
Date: 3/2017
DOI 10.17223/20710410/35/2

Keywords

число серий, цепь Маркова, расстояние по вариации, метод Чена-Стейна, смешанное распределение Пуассона, предельная теорема Пуассона, центральная предельная теорема, скрытая марковская модель, number of runs, Markov chain, total variation distance, Chen-Stein method, mixed Poisson distribution, Poisson limit theorem, normal limit theorem, hidden Markov model

Authors

Mezhennaya N. M.

Bauman Moscow State Technical University

natalia.mezhennaya@gmail.com

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Estimator for the distribution of the numbers of runs in a random sequence controlled by stationary Markov chain | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2017. № 35. DOI: 10.17223/20710410/35/2

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