Stationary distributions of perturbed Markov chains

2002

In an earlier paper (Hunter, 2002) it was shown that mean first passage times play an important role in determining bounds on the relative and absolute differences between the stationary probabilities in perturbed finite irreducible discrete time Markov chains. Further when two perturbations of the transition probabilities in a single row are carried out the differences between the stationary probabilities in the unperturbed and perturbed situations are easily expressed in terms of a reduced number of mean first passage times. Using this procedure we provide an updating procedure for mean first passage times to determine changes in the stationary distributions under successive perturbations. Simple procedures for determining both stationary distributions and mean first passage times in a finite irreducible Markov chain are also given. The techniques used in the paper are based upon the application of generalized matrix inverses.

Advances in Applied Probability, 1999

We consider a singularly perturbed ( nite state) Markov chain and provide a complete characterization of the fundamental matrix. In particular, we obtain a formula for the regular part simpler than a previous Schweitzer's formula, and the singular part is obtained via a reduction process similar to Delebecque's reduction for the stationary distribution. In contrast to previous approaches, one works with aggregate Markov chains of much smaller dimension than the original chain, an essential feature for practical computation. An application to mean rst-passage times is also presented.

Linear Algebra and its Applications, 2016

Computational procedures for the stationary probability distribution, the group inverse of the Markovian kernel and the mean first passage times of a finite irreducible Markov chain, are developed using perturbations. The derivation of these expressions involves the solution of systems of linear equations and, structurally, inevitably the inverses of matrices. By using a perturbation technique, starting from a simple base where no such derivations are formally required, we update a sequence of matrices, formed by linking the solution procedures via generalized matrix inverses and utilising matrix and vector multiplications. Four different algorithms are given, some modifications are discussed, and numerical comparisons made using a test example. The derivations are based upon the ideas outlined in Hunter,

Electronic Journal of Linear Algebra, 2003

Let T be an irreducible stochastic matrix with stationary vector π T . The conditioning of π T under perturbation of T is discussed by providing an attainable upper bound on the absolute value of the derivative of each entry in π T with respect to a given perturbation matrix. Connections are made with an existing condition number for π T , and the results are applied to the class of Markov chains arising from a random walk on a tree.

Perturbation analysis of Markov chains provides bounds on the effect that a change in a Markov transition matrix has on the corresponding stationary distribution. This paper compares and analyzes bounds found in the literature for finite and denumerable Markov chains and introduces new bounds based on series expansions. We discuss a series of examples to illustrate the applicability and numerical efficiency of the various bounds. Specifically, we address the question on how the bounds developed for finite Markov chains behave as the size of the system grows. In addition, we provide for the first time an analysis of the relative error of these bounds. For the case of a scaled perturbation we show that perturbation bounds can be used to analyze stability of a stable Markov chain with respect to perturbation with an unstable chain.

2011

Questions are posed regarding the influence that the column sums of the transition probabilities of a stochastic matrix (with row sums all one) have on the stationary distribution, the mean first passage times and the Kemeny constant of the associated irreducible discrete time Markov chain. Some new relationships, including some inequalities, and partial answers to the questions, are given using a special generalized matrix inverse that has not previously been considered in the literature on Markov chains.

INFOR: Information Systems and Operational Research, 1997

In this paper, based on probabilistic arguments, we obtain an explicit solution of the stationary distribution for a discrete time Markov chain with an upper Hessenberg time stationary transition probability matrix. Our solution then leads to a numerically stable and efficient algorithm for computing stationary probabilities. Two other expressions for the stationary distribution are also derived, which lead to two alternative algorithms. Numerical analysis of the algorithms is given, which shows the reliability and efficiency of the algorithms. Examples of applications are provided, including results of a discrete time state dependent batch arrive queueing model. The idea used in this paper can be generalized to deal with Markov chains with a more general structure.

arXiv (Cornell University), 2014

We consider a class of discrete time Markov chains with state space [0, 1] and the following dynamics. At each time step, first the direction of the next transition is chosen at random with probability depending on the current location. Then the length of the jump is chosen independently as a random proportion of the distance to the respective end point of the unit interval, the distributions of the proportions being fixed for each of the two directions. Chains of that kind were subjects of a number of studies and are of interest for some applications. Under simple broad conditions, we establish the ergodicity of such Markov chains and then derive closed form expressions for the stationary densities of the chains when the proportions are beta distributed with the first parameter equal to 1. Examples demonstrating the range of stationary distributions for processes described by this model are given, and an application to a robot coverage algorithm is discussed.

Advances in Applied Probability, 2005

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 probabilit...

Advances in Applied Probability, 2005

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.