Asymptotic Behaviour of Stationary Distributions for Countable Markov Chains, with Some Applications

Iasnogorodski, Roudolf

doi:10.2307/3318715

Cited by 11 publications

(11 citation statements)

References 15 publications

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“…Next we prove the uniform exponential decay of ν N and ν. Again these results are closely related to existing results in the literature, such as those in [18,25,30], Chapters 6 and 7 of [11], Section 2.2 of [3], or Section 16.3 of [26].…”

Section: Proof Of Theorem 81 (A) and (B)supporting

confidence: 87%

Rank-Driven Markov Processes

Grinfeld

Knight²,

Wade³

2011

J Stat Phys

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We study a class of Markovian systems of N elements taking values in [0, 1] that evolve in discrete time t via randomized replacement rules based on the ranks of the elements. These rank-driven processes are inspired by variants of the Bak-Sneppen model of evolution, in which the system represents an evolutionary 'fitness landscape' and which is famous as a simple model displaying self-organized criticality. Our main results are concerned with long-time large-N asymptotics for the general model in which, at each time step, K randomly chosen elements are discarded and replaced by independent U [0, 1] variables, where the ranks of the elements to be replaced are chosen, independently at each time step, according to a distribution κ N on {1, 2, . . . , N } K . Our main results are that, under appropriate conditions on κ N , the system exhibits threshold behaviour at s * ∈ [0, 1], where s * is a function of κ N , and the marginal distribution of a randomly selected element converges to U [s * , 1] as t → ∞ and N → ∞. Of this class of models, results in the literature have previously been given for special cases only, namely the 'mean-field' or 'random neighbour' Bak-Sneppen model. Our proofs avoid the heuristic arguments of some of the previous work and use Foster-Lyapunov ideas. Our results extend existing results and establish their natural, more general context. We derive some more specialized results for the particular case where K = 2. One of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic Markov chains on increasing state-spaces, which may be of independent interest.

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Section: Proof Of Theorem 81 (A) and (B)supporting

confidence: 87%

Rank-Driven Markov Processes

Grinfeld

Knight²,

Wade³

2011

J Stat Phys

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“…In the zero-drift case with general (oblique) reflections, in the 1980s Varadhan and Williams [29] had showed that the process was well-defined, and then Williams [30] gave the recurrence classification, thus preceding the random walk results of [6,11], and, in the recurrent cases, asymptotics of stationary measures (cf. [4] for the discrete setting). Passage-time moments were later studied in [7,24], by providing a continuum version of the results of [5], and in [2], using discrete approximation [1].…”

Section: Related Literaturementioning

confidence: 99%

Reflecting random walks in curvilinear wedges

Menshikov,

Mijatović,

Wade

2020

Preprint

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We study a random walk (Markov chain) in an unbounded planar domain whose boundary is described by two curves of the form x 2 = a + x β + 1 and x 2 = −a − x β − 1 , with x 1 ≥ 0. In the interior of the domain, the random walk has zero drift and a given increment covariance matrix. From the vicinity of the upper and lower sections of the boundary, the walk drifts back into the interior at a given angle α + or α − to the relevant inwards-pointing normal vector. Here we focus on the case where α + and α − are equal but opposite, which includes the case of normal reflection. For 0 ≤ β + , β − < 1, we identify the phase transition between recurrence and transience, depending on the model parameters, and quantify recurrence via moments of passage times.

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“…(2) There is a vast literature on the tail asymptotics of SDs for DTMCs on the integers, cf. [7,15,16,20,29,37,38]. From the relationship between an SD of a CTMC and that of its embedded DTMC [40,42], one can readily estimate the tail of the SD of the DTMC based on Theorem 5.8, when both SDs exist.…”

Section: Quasi-stationary Distributionsmentioning

confidence: 99%

Dynamics of continuous time Markov chains with applications

Xu,

Hansen,

Wiuf

2019

Preprint

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This paper contributes an in-depth study of properties of continuous time Markov chains (CTMCs) on non-negative integer lattices, with particular interest in one-dimensional CTMCs with polynomial transitions rates. Such stochastic processes are abundant in applications, in particular within biology. We study the classification of states for general CTMCs on the non-negative integer lattices, by characterizing the set of absorbing states (similarly, trapping, escaping, positive irreducible components and quasi-irreducible components). For CTMCs on non-negative integers with polynomial transition rates, we provide threshold criteria in terms of easily computable parameters for various dynamical properties such as explosivity, recurrence, transience, positive/null recurrence, implosivity, and existence and non-existence of passage times. In particular, simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions are obtained. Moreover, an identity for stationary measures is established and tail asymptotics for stationary distributions is given. A similar identity as well as asymptotics is derived for quasi-stationary distributions. Finally, we apply our results to stochastic reaction networks.

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Asymptotic Behaviour of Stationary Distributions for Countable Markov Chains, with Some Applications

Cited by 11 publications

References 15 publications

Rank-Driven Markov Processes

Rank-Driven Markov Processes

Reflecting random walks in curvilinear wedges

Dynamics of continuous time Markov chains with applications

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