Sensitivity of finite markov chains under perturbation. This paper compares and analyzes bounds found in the literature for finite and denumerable markov chains and introduces new bounds based. This paper is devoted to perturbation analysis of denumerable markov chains. Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain, where. Seneta school of mathematics and statistics, university of sydney, nsu. The new perturbation bounds are applicable to a wide range of problems. Pdf series expansions for continuoustime markov chains. The previously developed new perturbationiteration algorithm has been applied to differential equation systems for the first time. Perturbation bounds for structured robust stability.
Perturbation analysis for continuoustime markov chains. Two approaches to the construction of perturbation bounds. Bounds on convergence of entropy rate approximations in. We use information technology and tools to increase productivity and facilitate new forms of scholarship. Perturbation results for nearly uncoupled markov chains with applications to iterative methods. New perturbation bounds for denumerable markov chains core. June 29, 2018 abstract in this paper, new conditions for the stability of vgeometrically ergodic markov chains are introduced. Australia received september 1992 revised november 1992 abstract. A critical feature of the technique is a middle step that breaks the problem into solvable and perturbation parts. Perturbation bounds for quantum markov processes and their. The problems of stability and the corresponding estimates were considered for new classes of processes in zeifman. Other applications of our results to phasetype queues will be. I present a perturbative approach that allows one to uniformly bound the difference between the hitting time moment generating functions of two markov chains in a neighbourhood of the origin. Comparison of perturbation bounds for the stationary distribution of a markov chain.
Perturbation theory and finite markov chains researchgate. Xiaoyue li a,1, rui wang a,b, george yin c,2 a school of mathematics and statistics, northeast normal university, changchun, jilin, 024, china department of economics, university of kansas, lawrence, ks 66045, usa c department of. Markov chains, deviation matrix, linear pogramming, perturbation matrix analysis 1. The algorithm is tested for a single equation, coupled two equations, and coupled three equations. Perturbation bounds for markov chains with general state. This is useful for studying how sensitive the original systems eigenvectors and eigenvalues are to changes in the system.
Numerical examples are given to illustrate the performance of the algorithm. If p is a transition matrix of a markov chain, andequation is derived by perturbing. Strong stability and perturbation bounds for discrete markov chains. Journal of science, engineering and technology, waset world academy of science, engineering and technology ed pp. Perturbation analysis of finite markov chains has received much attention in the literature over recent years see in. New perturbation bounds for denumerable markov chains, linear algebra and its applications, 432, 16271649.
Qbd processes, which constitute a wide class of structured markov chains. In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system that is perturbed from one with known eigenvectors and eigenvalues. Also, we obtain perturbation bounds with respect to different quantities. Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain, where the bounds are given by the. D and d are derived in terms of a drift condition, where.
A critical account of perturbation analysis of markov chains. Perturbation theory for markov reward processes with applications to queueing systems volume 20 issue 1 nico m. We study general statespace markov chains that depend on a parameter, say, sufficient conditions are established for the stationary performance of such a markov chain to be differentiable with respect to specifically, we study the case of unbounded performance functions and thereby extend the result on weak differentiability of stationary distributions of markov chains to unbounded. We establish upper and lower bounds on this condition number in terms of subdominant eigenvalues of the transition map. Under suitable stability conditions, numerical approximations can be derived from the update formulas, and we show that the algorithms converge at a geometric. New perturbation bounds for denumerable markov chains citeseerx. This thesis is concerned with studying the hitting time of an absorbing state on markov chain models that have a countable state space.
Nunezqueijaperturbation analysis for denumerable markov chains with. This article provides series expansions of the stationary distribution of a finite markov chain. This section may be regarded as a complement of daleys work 3. Strong stability and perturbation bounds for discrete markov. New perturbation bounds for denumerable markov chains. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Perturbation theory for markov reward processes with. Mar 15, 2010 new perturbation bounds for denumerable markov chains new perturbation bounds for denumerable markov chains mouhoubi, zahir.
On the existence of quasistationary distributions in. Regular perturbation of vgeometrically ergodic markov chains. Moreover, we obtain perturbation bounds on the stationary distributions, which extends the results by liu 2012 for. In the present paper we propose an approach to the construction of general estimates for the perturbation bounds of markov chains in terms of special weighted norms related to total variation.
Strong stability and perturbation bounds for discrete. Puterman skip to main content we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Dec 19, 2017 the aim of this paper is to investigate the stability of markov chains with general state space. Apr 15, 2008 read strong stability and perturbation bounds for discrete markov chains, linear algebra and its applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Markovswitching dsge msdsge modeling has become a growing body of literature on economic and policy issues related to structural shifts. Harcourt bruce jovanovich, publishers boston san diego new york london sydney tokyo toronto. Sensitivity of finite markov chains under perturbation e. Our bounds are sharp, we do not impose any structural assumptions on the transition matrix or on the perturbation, and computing the bounds has the same complexity as computing the invariant distribution or computing other bounds in the literature. On perturbation bounds for continuoustime markov chains. Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain, where the bounds are given by the weighted. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of inhomogeneous markov chains with continuous time and a finite or countable state space. New perturbation bounds for denumerable markov chains new perturbation bounds for denumerable markov chains mouhoubi, zahir.
Perturbations of countable markov chains and processes. We study the parametric perturbation of markov chains with denumerable state spaces. Haverford college 2005 dissertation submitted in partial satisfaction of. Gaussian elimination, perturbation theory and markov. Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain, where the bounds are given by the weighted supremum norm. Bounds on convergence of entropy rate approximations in hidden markov processes by nicholas f. To address the effects of uncertainty in probability estimates, in previous work we have developed a variety of techniques for perturbation analysis of discrete and continuoustime markov chains dtmcs and ctmcs. We consider three basic matrix norms to capture the perturbation distance, and focus on the computational aspect. Linear algebra and its applications journal homepage. This leads to an efficient numerical algorithm for computing the stationary distribution of a finite markov chain. Solutions are compared with those of variational iteration method and numerical solutions, and a good.
Semigroups of conditioned shifts and approximation of markov processes kurtz, thomas g. Perturbation analysis for denumerable markov chains with application to queueing models. Perturbation analysis for denumerable markov chains with. We present update formulas that allow us to express the stationary distribution of a continuoustime markov process with denumerable state space having generator matrix q through a continuoustime markov process with generator matrix q. We investigate perturbation for continuoustime markov chains ctmcs on a countable state space. Let p be the transition matrix of a positive recurrent markov chain on the integers. Series expansions for finitestate markov chains semantic. In this paper, our interest is in the perturbation analysis of level. Jul 17, 2006 2011 perturbation analysis of continuoustime absorbing markov chains.
Apr 30, 2015 we investigate perturbation for continuoustime markov chains ctmcs on a countable state space. Two approaches to the construction of perturbation bounds for. Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain, where the. Abstractthis paper is devoted to perturbation analysis of denumerable markov chains. Measurevalued differentiation for stationary markov chains.
Siam journal on numerical analysis volume 25, issue 3. In the present paper we propose an approach to the construction of general estimates for the perturbation bounds of markov chains in terms of special weighted. Sharp entrywise perturbation bounds for markov chains. Finally, in section 4, we explicitly obtain the quasistationary distributions of a leftcontinuous random walk to demonstrate the usefulness of our results. Perturbationiteration method for firstorder differential. This paper develops a general perturbation methodology for constructing highorder approximations to the solutions of msdsge models. We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Series expansions for continuoustime markov processes. Siam journal on numerical analysis society for industrial.
We study the parametric perturbation of markov chains with denumerable. An example in denumerable decision processes fisher, lloyd and ross, sheldon m. Perturbation methods for markovswitching dsge models. For finite irreducible markov chains many perturbation bounds for the. The aim of this paper is to investigate the stability of markov chains with general state space. Moment bounds and ergodicity of switching diffusion systems involving twotimescale markov chains. We show how to reduce the complex markovswitching problem to solving a system of quadratic polynomial equations. Summary in this paper, our interest is in the perturbation analysis of level. Twodimensional harmonic oscilator 3 timedependent perturbation theory 4 literature igor luka cevi c. However, there are only few references available on perturbation analysis of markov chains with an in. Mar 19, 20 we investigate the stability of quantum markov processes with respect to perturbations of their transition maps.
Gaussian elimination, perturbation theory, and markov chains g. For many models it is challenging to study the hitting time directly. Regular perturbation of vgeometrically ergodic markov chains deborah ferre, loic herve, james ledoux. Bounds are provided for the deviation between the stationary distribution of the perturbed and nominal. We perform perturbation analysis in the setting of discretetime markov chains. Meyer 1992 has developed inequalities in terms of the nonunit eigenvalues h, j 2.
Taylor series expansions for stationary markov chains. Reliability modelling and data analysis of vacuum circuit breaker subject to random shocks. Robust stability of a linear multivariable system, in the sense of robustness under multiplicative transfer function perturbation, is necessarily preserved under sufficiently small perturbations in the model parameters i. Finite continuous time markov chains theory of probability. Lower bounds, which show that the individual perturbation bounds are rateoptimal, are also given. Create an aipowered research feed to stay up to date with new papers like this posted to arxiv. Measurevalued differentiation for stationary markov. Moment bounds and ergodicity of switching diffusion systems. Timedependent perturbation theory literature 1 timeindependent nondegenerate perturbation theory general formulation firstorder theory secondorder theory 2 timeindependent degenerate perturbation theory general formulation example.
Introduction a perturbation in a markov chain can be referred as a slight change in the entries of the corresponding transition stochastic matrix, resulting in structural changes in the underlying process, for. Additive perturbation bounds on the eigenvectors of a hermitian matrix. Perturbation methods for markovswitching models andrew foerstery juan rubioramirezz dan waggonerx tao zhaaugust 2, 2012 abstract markov switching models are a way to consider discrete changes in the economic environment, such as policy changes, and allow agents in the economy to form expectations over these changes. Perturbation bounds for markov chains with general state space. Strong stability and perturbation bounds for discrete markov chains strong stability and perturbation bounds for discrete markov chains rabta, boualem. We study general statespace markov chains that depend on a parameter, say, sufficient conditions are established for the stationary performance of such a markov chain to be differentiable with respect to. Error bounds for augmented truncation approximations of markov. New perturbation bounds for denumerable markov chains linear algebra and its applications, vol. Introduction a perturbation in a markov chain can be referred as a slight change in the entries of the corresponding transition stochastic matrix, resulting in structural changes in the underlying process, for example, sets. Rateoptimal perturbation bounds for singular subspaces with. Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. Singular perturbation analysis for countable markov chains.
We investigate the stability of quantum markov processes with respect to perturbations of their transition maps. We consider both regular and singular perturbations. Perturbation bounds perturbation analysis of markov chains residual matrix norm ergodicity coef. Discretetime markov chains with twotime scales and a countable. New perturbation bounds for denumerable markov chains, linear. For finite irreducible markov chains many perturbation bounds for the stationary vector are available. By the latter we mean that transition probabilities of a markov chain, with several ergodic classes, are perturbed such that rare transitions among the different ergodic classes of the unperturbed chain are allowed. Qbd processes, which constitute a wide class of structured markov. Denumerable markov chains can be used to represent many real systems. Rateoptimal perturbation bounds for singular subspaces. Probabilistic model checking of perturbed mdps with. On the existence of quasistationary distributions in denumerable rtransient markov chains authors. Singularly perturbed discretetime markov chains, siam journal on applied mathematics, 6, 834854. We provide a unified approach to pamc for finite and denumerable markov.
We present new conditions for the strong stability of markov chains after a small perturbation of their transition kernels. Introduction to stochastic processes, prenticehall, new jersey. The iteration algorithm for systems is developed first. Introduction the purpose of this paper is to describe the special problems that emerge when gaussian elimination is used to determinine the stead.