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Prikl. Diskr. Mat., 2017, Number 35, Pages 14–28 (Mi pdm575)  

Theoretical Foundations of Applied Discrete Mathematics

Estimator for the distribution of the numbers of runs in a random sequence controlled by stationary Markov chain

N. M. Mezhennaya

Bauman Moscow State Technical University

Abstract: The sequences of random characters from a finite set $\mathcal A$ 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 $\rho_{TV}$ between the distribution of a random vector $\varsigma_\mathcal A$ with components being numbers of runs in a controlled sequence of an enough length $T$ and accompanying multidimensional Poisson distribution $\mathrm{Pois}(\lambda_\mathcal A)$. The estimate is $\rho_{TV}(\mathcal L(\varsigma_\mathcal A),\mathrm{Pois}(\lambda_\mathcal A))\leq\gamma(\gamma T(p^*)^{s_*}+1)$, where $\gamma^2=|\mathcal A|^2(2s^*+3)(p^*)^{s_*}$, $s_*$ ($s^*$) is the minimum (maximum) length of run in the set of components of the vector $\varsigma_\mathcal A$, and $p^*$ is the maximum character probability in distributions given on $\mathcal A$. 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 $\rho_{TV}$, we deduce the multidimensional Poisson and normal limit theorems for the random vector $\varsigma_\mathcal A$ under appropriate conditions for scheme parameters.

Keywords: number of runs, Markov chain, total variation distance, Chen–Stein method, mixed Poisson distribution, Poisson limit theorem, normal limit theorem, hidden Markov model.

DOI: https://doi.org/10.17223/20710410/35/2

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Document Type: Article
UDC: 519.214.5

Citation: N. M. Mezhennaya, “Estimator for the distribution of the numbers of runs in a random sequence controlled by stationary Markov chain”, Prikl. Diskr. Mat., 2017, no. 35, 14–28

Citation in format AMSBIB
\Bibitem{Mez17}
\by N.~M.~Mezhennaya
\paper Estimator for the distribution of the numbers of runs in a~random sequence controlled by stationary Markov chain
\jour Prikl. Diskr. Mat.
\yr 2017
\issue 35
\pages 14--28
\mathnet{http://mi.mathnet.ru/pdm575}
\crossref{https://doi.org/10.17223/20710410/35/2}


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