Asymptotics for Quasi-stationary Distributions of Perturbed Discrete Time Semi-Markov Processes
In this paper we study quasi-stationary distributions of non-linearly perturbed semi-Markov processes in discrete time. This type of distributions are of interest for analysis of stochastic systems which have finite lifetimes but are expected to persist f
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Abstract In this paper we study quasi-stationary distributions of non-linearly perturbed semi-Markov processes in discrete time. This type of distributions are of interest for analysis of stochastic systems which have finite lifetimes but are expected to persist for a long time. We obtain asymptotic power series expansions for quasistationary distributions and it is shown how the coefficients in these expansions can be computed from a recursive algorithm. As an illustration of this algorithm, we present a numerical example for a discrete time Markov chain. Keywords Semi-Markov process · Perturbation · Quasi-stationary distribution Asymptotic expansion · Renewal equation · Markov chain
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1 Introduction This paper is a sequel of [22] where recursive algorithms for computing asymptotic expansions of moment functionals for non-linearly perturbed semi-Markov processes in discrete time are presented. Here, these expansions play a fundamental role for constructing asymptotic expansions of quasi-stationary distributions for such processes. Let us remark that all notation, conditions, and key results which we need here are repeated. However, some extensive formulas needed for computation of coefficients in certain asymptotic expansions are not repeated. Thus, the present paper is essentially self-contained. Quasi-stationary distributions are useful for studies of stochastic systems with random lifetimes. Usually, for such systems, the evolution of some quantity of interest is described by some stochastic process and the lifetime of the system is the first time this process hits some absorbing subset of the state space. For such processes, the stationary distribution will be concentrated on this absorbing subset. However, if we expect that the system will persist for a long time, the stationary distribution may M. Petersson (B) Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden e-mail: [email protected] © Springer International Publishing Switzerland 2016 S. Silvestrov and M. Ranˇci´c (eds.), Engineering Mathematics II, Springer Proceedings in Mathematics & Statistics 179, DOI 10.1007/978-3-319-42105-6_9
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not be an appropriate measure for describing the long time behaviour of the process. Instead, it might be more relevant to consider so-called quasi-stationary distributions. This type of distributions is obtained by taking limits of transition probabilities which are conditioned on the event that the process has not yet been absorbed. Models of the type described above arise in many areas of applications such as epidemics, genetics, population dynamics, queuing theory, reliability, and risk theory. For example, in population dynamics models the number of individuals may be modelled by some stochastic process and we can consider the extinction time of the population as the lifetime. In epidemic models, the process may describe the evolution of the number of infected individuals and we can regard the end of the epidemic as the lifetime. We consider, for every ε ≥ 0, a dis
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