Central limit theorem for an additive functional of a Markov process, stable in the Wesserstein metric

Walczuk, Anna

doi:10.2478/v10062-008-0016-0

Cited by 6 publications

(10 citation statements)

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“…for continuous time Markov processes whose state space is allowed to be non-compact and an observable that may be unbounded (we only require it to be Lipschitz). In [34] Markov processes stable in the Wasserstein type metric, stronger than the one considered here, have been examined and an analogue of Theorem 2.1 has been shown. After finishing this manuscript we have learned about the results of [36], where the central limit theorem for solutions of Navier-Stokes equations has been studied.…”

Section: Introductionmentioning

confidence: 98%

“…We stress that the processes considered in Theorem 2.1 presented below need not be stationary. In this context the results of [18,22,36] and [34] should be mentioned. In Theorem 19.1.1 of [22] the central limit theorem is proved for every starting point of a Markov chain that is stable in the total variation metric (this is equivalent with the uniform mixing property of the chain).…”

Section: Introductionmentioning

confidence: 99%

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Central limit theorem for Markov processes with spectral gap in the Wasserstein metric

Komorowski

Walczuk

2012

Stochastic Processes and their Applications

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Suppose that {X t , t ≥ 0} is a non-stationary Markov process, taking values in a Polish metric space E. We prove the law of large numbers and central limit theorem for an additive functional of the form T 0 ψ(X s )ds, provided that the dual transition probability semigroup, defined on measures, is strongly contractive in an appropriate Wasserstein metric. Function ψ is assumed to be Lipschitz on E.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Central limit theorem for Markov processes with spectral gap in the Wasserstein metric

Komorowski

Walczuk

2012

Stochastic Processes and their Applications

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“…Thus, all assumptions of Theorem A.1 [40] are satisfied. By the Central Limit Theorem for martingales,…”

Section: Proof Of Theoremmentioning

confidence: 87%

The Central Limit Theorem for Random Dynamical Systems

Horbacz

2016

J Stat Phys

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“…So far, there are a few of papers on LLN and CLT for stochastic dynamical systems which are weakly ergodic; see e.g. [21,22,24,27]. In particular, the reference function f in [21,27] is assumed to be (bounded) Lipschitz with respect to a metric and the weak LLN is investigated; In [22], the LLN is established under some additional technical conditions (see [22,Theorem 5.1.10] for more details).…”

mentioning

confidence: 99%

“…[21,22,24,27]. In particular, the reference function f in [21,27] is assumed to be (bounded) Lipschitz with respect to a metric and the weak LLN is investigated; In [22], the LLN is established under some additional technical conditions (see [22,Theorem 5.1.10] for more details). In this paper, we will show that limit theorems established in [24] for uniformly mixing Markov processes apply well to the present model, where the observable f is Lipschitz continuous with respect to a quasi-metric.…”

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confidence: 99%