scholarly journals Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type

2005 ◽  
Vol 37 (4) ◽  
pp. 1075-1093 ◽  
Author(s):  
Quan-Lin Li ◽  
Yiqiang Q. Zhao
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Analysis ◽  
Positive Solution ◽  
Generating Function ◽  
Spectral Properties ◽  
Stationary Probability ◽  
Probability Vector ◽  
Stationary Probability Vector

In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by the R-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. The RG-factorization of both the repeating row and the Wiener-Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probability vector.

scholarly journals Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type

2005 ◽  
Vol 37 (04) ◽  
pp. 1075-1093 ◽  
Author(s):  
Quan-Lin Li ◽  
Yiqiang Q. Zhao
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Analysis ◽  
Positive Solution ◽  
Generating Function ◽  
Spectral Properties ◽  
Stationary Probability ◽  
Probability Vector ◽  
Stationary Probability Vector

In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by theR-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. TheRG-factorization of both the repeating row and the Wiener-Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probability vector.

Geometric bounds on iterative approximations for nearly completely decomposable Markov chains

1990 ◽  
Vol 27 (03) ◽  
pp. 521-529 ◽  
Author(s):  
Guy Louchard ◽  
Guy Latouche
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Perturbation Analysis ◽  
Stochastic Matrix ◽  
Stationary Probability ◽  
Probability Vector ◽  
Finite Markov Chain ◽  
Stationary Probability Vector

We consider a finite Markov chain with nearly-completely decomposable stochastic matrix. We determine bounds for the error, when the stationary probability vector is approximated via a perturbation analysis.

Geometric bounds on iterative approximations for nearly completely decomposable Markov chains

1990 ◽  
Vol 27 (3) ◽  
pp. 521-529 ◽  
Author(s):  
Guy Louchard ◽  
Guy Latouche
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Perturbation Analysis ◽  
Stochastic Matrix ◽  
Stationary Probability ◽  
Probability Vector ◽  
Finite Markov Chain ◽  
Stationary Probability Vector

We consider a finite Markov chain with nearly-completely decomposable stochastic matrix. We determine bounds for the error, when the stationary probability vector is approximated via a perturbation analysis.

scholarly journals A New GMRES(m) Method for Markov Chains

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Bing-Yuan Pu ◽  
Ting-Zhu Huang ◽  
Chun Wen
Keyword(s):  
Markov Chain ◽  
Convergence Rate ◽  
Numerical Experiments ◽  
Gmres Method ◽  
Stationary Probability ◽  
Probability Vector ◽  
Vector Extrapolation ◽  
Restarted Gmres ◽  
Stationary Probability Vector ◽  
Irreducible Markov Chain

This paper presents a class of new accelerated restarted GMRES method for calculating the stationary probability vector of an irreducible Markov chain. We focus on the mechanism of this new hybrid method by showing how to periodically combine the GMRES and vector extrapolation method into a much efficient one for improving the convergence rate in Markov chain problems. Numerical experiments are carried out to demonstrate the efficiency of our new algorithm on several typical Markov chain problems.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

A New Algorithm for Computing the Ergodic Probability Vector for Large Markov Chains

1990 ◽  
Vol 4 (1) ◽  
pp. 89-116 ◽  
Author(s):  
Ushlo Sumita ◽  
Maria Rieders
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
The State ◽  
Special Structure ◽  
Probability Vector ◽  
Rank Reduction ◽  
Replacement Process ◽  
Novel Algorithm

A novel algorithm is developed which computes the ergodic probability vector for large Markov chains. Decomposing the state space into lumps, the algorithm generates a replacement process on each lump, where any exit from a lump is instantaneously replaced at some state in that lump. The replacement distributions are constructed recursively in such a way that, in the limit, the ergodic probability vector for a replacement process on one lump will be proportional to the ergodic probability vector of the original Markov chain restricted to that lump. Inverse matrices computed in the algorithm are of size (M – 1), where M is the number of lumps, thereby providing a substantial rank reduction. When a special structure is present, the procedure for generating the replacement distributions can be simplified. The relevance of the new algorithm to the aggregation-disaggregation algorithm of Takahashi [29] is also discussed.

scholarly journals Heavy-tailed asymptotics of stationary probability vectors of Markov chains of gi/g/1 type

2005 ◽  
Vol 37 (02) ◽  
pp. 482-509 ◽  
Author(s):  
Quan-Lin Li ◽  
Yiqiang Q. Zhao
Keyword(s):  
Transition Matrix ◽  
Stationary Probability ◽  
Probability Vector ◽  
Necessary And Sufficient Condition ◽  
Hopf Equation ◽  
Novel Approach ◽  
Matrix Sequence ◽  
Heavy Tailed ◽  
Necessary And Sufficient ◽  
Stationary Probability Vector

In this paper, we provide a novel approach to studying the heavy-tailed asymptotics of the stationary probability vector of a Markov chain of GI/G/1 type, whose transition matrix is constructed from two matrix sequences referred to as a boundary matrix sequence and a repeating matrix sequence, respectively. We first provide a necessary and sufficient condition under which the stationary probability vector is heavy tailed. Then we derive the long-tailed asymptotics of the R-measure in terms of the RG-factorization of the repeating matrix sequence, and a Wiener-Hopf equation for the boundary matrix sequence. Based on this, we are able to provide a detailed analysis of the subexponential asymptotics of the stationary probability vector.

scholarly journals The role of the range of dispersal in a nonlocal Fisher-KPP equation: an asymptotic analysis

2020 ◽  
pp. 2050032
Author(s):  
Julien Brasseur
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Positive Solution ◽  
Nonlocal Problem ◽  
Type Equation ◽  
Independent Interest ◽  
Kpp Equation ◽  
Open Question

In this paper, we study the asymptotic behavior as [Formula: see text] of solutions [Formula: see text] to the nonlocal stationary Fisher-KPP type equation [Formula: see text] where [Formula: see text] and [Formula: see text]. Under rather mild assumptions and using very little technology, we prove that there exists one and only one positive solution [Formula: see text] and that [Formula: see text] as [Formula: see text] where [Formula: see text]. This generalizes the previously known results and answers an open question raised by Berestycki et al. Our method of proof is also of independent interest as it shows how to reduce this nonlocal problem to a local one. The sharpness of our assumptions is also briefly discussed.

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