A unified perturbation analysis framework for countable Markov chains

Jiang, Shuxia; Liu, Yuanyuan; Tang, Yingchun

doi:10.1016/j.laa.2017.05.002

Cited by 11 publications

(5 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recent advances in this direction can be found in [1], [2], [19], and [31]. The Vnormwise perturbation bounds for CTMCs have also received recent attention; see [9], [12], and [20]. The bounds on the solution of the Poisson equation were presented in [8] in terms of drift conditions, which were further applied to the perturbation analysis in the V -norm.…”

Section: Introductionmentioning

confidence: 99%

Error bounds for augmented truncation approximations of Markov chains via the perturbation method

Liu

2018

Adv. Appl. Probab.

Self Cite

View full text Add to dashboard Cite

LetPbe the transition matrix of a positive recurrent Markov chain on the integers with invariant probability vectorπT, and let(n)P̃ be a stochastic matrix, formed by augmenting the entries of the (n+ 1) x (n+ 1) northwest corner truncation ofParbitrarily, with invariant probability vector(n)πT. We derive computableV-norm bounds on the error betweenπTand(n)πTin terms of the perturbation method from three different aspects: the Poisson equation, the residual matrix, and the norm ergodicity coefficient, which we prove to be effective by showing that they converge to 0 asntends to ∞ under suitable conditions. We illustrate our results through several examples. Comparing our error bounds with the ones of Tweedie (1998), we see that our bounds are more applicable and accurate. Moreover, we also consider possible extensions of our results to continuous-time Markov chains.

show abstract

Section: Introductionmentioning

confidence: 99%

Error bounds for augmented truncation approximations of Markov chains via the perturbation method

Liu

2018

Adv. Appl. Probab.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Then the Markov chain X(t) is 1D-exponentially weakly ergodic, under perturbations small enough ( 16) the perturbed chain X(t) is also 1D-exponentially weakly ergodic and perturbation bound (38) in the 1D-norm holds. If, moreover, W = inf i≥1 d i i > 0, then both chains X(t) and X(t) have limit expectations and estimate (18) holds for the perturbation of the mathematical expectation.…”

Section: Convergence Rate Estimates and Perturbation Bounds For Main ...mentioning

confidence: 99%

“…Following the ideas of N. V. Kartashov (see a detailed description in [23]), most authors use the probability methods to study ergodicity and perturbation bounds of stationary chains (with a finite, countable or general state space) in various norms [3,12,35]. For a wide class of (mainly) stationary discrete-time chains a close approach was considered in [29] and more recent papers [1,18,25,26,28,36,38,44,45,46,70].…”

Section: Introductionmentioning

confidence: 99%

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

2020

View full text Add to dashboard Cite

The paper is largely of a review nature. It considers two main methods used to study stability and obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible "perturbed"processes are calculated.

show abstract

“…In contrast, singular perturbations correspond to cases where some state transitions in a Markov chain are considerably faster than others, so we could think of "large-magnitude perturbations" or multiple time scales. While the typical approach to singular perturbations centers on asymptotic expansions [25,26,28], perturbation-bound approaches to singular perturbations have also been developed [33,34]. Thus, some of the results that we discuss could, in principle, be applied to singular-perturbation problems.…”

Section: Introductionmentioning

confidence: 99%

The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective

Mitrophanov

2024

Mathematics

View full text Add to dashboard Cite

Perturbation bounds are powerful tools for investigating the phenomenon of insensitivity to perturbations, also referred to as stability, for stochastic and deterministic systems. This perspective article presents a focused account of some of the main concepts and results in inequality-based perturbation theory for finite state-space, time-homogeneous, continuous-time Markov chains. The diversity of perturbation bounds and the logical relationships between them highlight the essential stability properties and factors for this class of stochastic processes. We discuss the linear time dependence of general perturbation bounds for Markov chains, as well as time-independent (i.e., time-uniform) perturbation bounds for chains whose stationary distribution is unique. Moreover, we prove some new results characterizing the absolute and relative tightness of time-uniform perturbation bounds. Specifically, we show that, in some of them, an equality is achieved. Furthermore, we analytically compare two types of time-uniform bounds known from the literature. Possibilities for generalizing Markov-chain stability results, as well as connections with stability analysis for other systems and processes, are also discussed.

show abstract

A unified perturbation analysis framework for countable Markov chains

Cited by 11 publications

References 23 publications

Error bounds for augmented truncation approximations of Markov chains via the perturbation method

Error bounds for augmented truncation approximations of Markov chains via the perturbation method

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective

Contact Info

Product

Resources

About