Asymptotic Expansions for Stationary Distributions of Nonlinearly Perturbed Semi-Markov Processes

Abstract

Asymptotic expansions with explicit upper bounds for remainders are given for stationary distributions of nonlinearly perturbed semi-Markov processes with finite phase spaces. The corresponding algorithms are based on a special technique of sequential phase space reduction, which can be applied to processes with an arbitrary asymptotic communicative structure of phase spaces.

Nonlinearly Perturbed Birth-Death-Type Models

“…The present paper is devoted to studies of asymptotic expansions for stationary and conditional quasi-stationary distributions for perturbed birthdeath-type semi-Markov processes. The algorithms of sequential phase space reduction for perturbed semi-Markov processes combined with techniques of Laurent asymptotic expansions developed in the recent papers by Silvestrov, D. and Silvestrov, S. (2015Silvestrov, S. ( , 2016 are applied to birth-death-type semi-Markov processes. In this model, the proposed algorithms of phase space reduction preserve the birth-death structure for reduced semi-Markov processes.…”

“…The model in (51)-(52) is an SIS-epidemic, since infected individuals become susceptible after recovery. It is essentially a special case of (38)-(39), with θ 1 = θ 2 = θ 3 = 1, α 1 = 1 and α 2 = 0, although immigration is parametrized differently in (38) and (51). Assume that the external contact rate…”

“…The results of the present section are based on the explicit formula (104) for expected return times and the expressions which connect these quantities with stationary and conditional quasi-stationary distributions. We obtain the first and second order asymptotic expansions from these formulas by using operational rules for Laurent asymptotic expansions presented in Silvestrov, D. and Silvestrov, S. (2015Silvestrov, S. ( , 2016. Some of these operational rules which are relevant for this paper can be found in Subsection 6.1.…”

Asymptotic Expansions for Stationary Distributions of Nonlinearly Perturbed Semi-Markov Processes. 2

“…We would like to conclude the introduction with the remark that the present paper is a shorten version of the report Silvestrov and Silvestrov (2016b), where one can find some additional details of proofs, comments and references.…”

Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes

“…Alternatively, the methods based on regenerative properties of Markov chains and semi-Markov processes, in particular, relations which link stationary probabilities and expectations of return times have been used for getting approximations for expectations of hitting times and stationary distributions in works by Grassman, Taksar and Heyman (1985), Hassin and Haviv (1992) and Hunter (2005). Also, the above mentioned relations and methods based on asymptotic expansions for nonlinearly perturbed regenerative processes developed in works by Silvestrov (1995Silvestrov ( , 2007Silvestrov ( , 2010, Englund and Silvestrov (1997), Gyllenberg and Silvestrov (1998, 1999a, 2008, Englund (2000Englund ( , 2001, Ni, Silvestrov and Malyarenko (2008), Ni (2010aNi ( , b, 2011Ni ( , 2012Ni ( , 2014, Silvestrov and Petersson (2013) and Petersson (2013aPetersson ( , b, 2014 have been used for getting asymptotic expansions for stationary and quasi-stationary distributions for nonlinearly perturbed Markov chains and semi-Markov processes with absorption.…”