Институт вычислительной математики
и математической геофизики

The International Conference on Computational Mathematics


Statistic modeling and Monte Carlo methods

The Numerical Characteristics in Stochastic Sequences

Balakin S.V.

Novosibirsk State University (Novosibirsk)

documentclass[12pt]{article} usepackage[cp866]{inputenc} usepackage[T2A]{fontenc} usepackage[english]{babel} usepackage{indentfirst} extwidth 150.0mm oddsidemargin 4.8mm evensidemargin 4.8mm itle{The Numerical Characteristics In Stochastic Sequences.} author{S. Balakin} egin{document} maketitle paragraph{1.} Joint distributions of different runs characteristics in binary Markov sequence are viewed. Also the first moments of these distributions are considered. There are joint and marginal distributions of the number of successes, the number of success runs and their first moments described in the report. For expectations, variances and covariances there are precise and asymptotic formulas with estimations. Formulas for normal approximations are derived. The obtained results have different applications in genetics, simulation of nature processes, economics and statistics. paragraph{2.} In particular, consider the binary Markov sequence $xi$ of random variables $xi(k)$, $k ge 0$ with the value set $C={1,0}$, initial vector $A$ and transition matrix $Q$: [ egin{array}{cc} A=(a, 1 - a), & Q= left(egin{array}{cc} p & 1-p 1-q & q end{array} ight). end{array} ] Random variables equal to the number of successes and the number of success runs in the sequence $xi$ on the segment $[0,n]$ are correspondingly described by the equalities: [ egin{array}{cc} x(n) = xi (0) + sumlimits_{k = 1}^n {xi (k)}, & y(n) = xi(0) + sumlimits_{k = 1}^n {left( 1 - xi (k - 1) ight)} xi (k). end{array} ] paragraph{3.} There is the equality for the joint distribution of the number of successes and the number of success runs: egin{equation} label{6} P(x(n)=i,y(n)=j) = bleft(j - 1,i - 1,1 - p ight) sum limits_{alpha ,eta } {chi _{alpha eta } bleft(j - alpha - eta ,n - i,1 - q ight)}, end{equation} where $alpha, eta in {0,1}$, [ egin{array}{c} b(k,m,x) = left(egin{array}{c} m k end{array} ight) x^k left(1-x ight)^{m - k}, chi _{alpha eta } = left(alpha a + (1 - alpha )(1 - a) ight)left(eta left(1 - q ight) + left(1 - eta ight)left(1 - p ight) ight). end{array} ] paragraph{4.} With the method of probability generating functions we derive the precise formulas for the expectations and variances of the reviewed characteristics. Also the precise formulas for covariance and correlation are received. In the asymmetric and symmetric $(p=q)$ cases the following asymptotic expressions for covariance are derived: [ egin{array}{cc} Cov(x(n),y(n)) = O(n), & Cov(x(n),y(n)) = O(1). end{array} ] And so for correlation in the symmetric case: [ K(x(n),y(n)) = Oleft(1/n ight). ] paragraph{5.} The equality ( ef{6}) for the joint distribution of the random variables $x(n)$, $y(n)$ let us obtain the expression for its normal approximation. It is simple enough. The formulas for the normal approximation derived in cite[теорема 1.1.12]{Kolch} are used. paragraph{6.} As a case of quite natural characteristics in Markov sequences we can also consider such random variables as the length of cover run (cite{SBK}) and the maximum length of success runs. As for the binary Markov sequences there are precise formulas convenient for analyzing the above-mentioned characteristics. The probability generating functions are derived. The next important property of Markov sequences is discovered: as distinct from the the sequences of independent random variables, the lengths of runs in Markov sequences do not always decrease with the growth of the number of events and the corresponding distributions may have a complex behavior. The conditions, which are necessary and sufficient for the increase of these distributions in Markov case, are obtained. egin{thebibliography}{19} egin{small} ibitem{Kolch} Колчин В.Ф. emph{Случайные графы.} Москва, Физматлит, 2000. ibitem{SBK} Савельев Л.Я., Балакин С.В., Хромов Б.В. Накрывающие серии в двоичных марковских последовательностях. emph{Дискретная математика}, том 15, вып. 1, 2003. С.~50-76. end{small} end{thebibliography} end{document}

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