An almost sure invariance principle for additive functionals of Markov chains

We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the oriented percolation universality class. If the influence of the random medium on the walk is small in space-time regions where the medium is typical, we obtain a law of large numbers and an averaged central limit theorem for the walk via a regeneration construction under suitable coarse-graining.Such random walks occur naturally as spatial embeddings of ancestral lineages in spatial population models with local regulation. We verify that our assumptions hold for logistic branching random walks when the population density is sufficiently high. Keywords: Random walk; dynamical random environment; oriented percolation; supercritical cluster; central limit theorem in random environment; logistic branching random walk. AMS MSC 2010: 60K37; 60J10; 82B43; 60K35. * M.B. and A.D. gratefully acknowledge support by DFG priority programme SPP 1590 Probabilistic structures in evolution through grants BI 1058/3-1 respectively Pf 672/6-1.

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“…From this representation the (functional) central limit theorem can be deduced; see e.g. Chapter 1 in [19] or Theorem 2 in [26].…”

Section: Setmentioning

confidence: 99%

Random walks in dynamic random environments and ancestry under local population regulation

Birkner¹,

Černý²,

Depperschmidt³

2016

Electron. J. Probab.

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“…of (33) are absolutely convergent in L 2 (µ), and therefore, by (28) and (48), the expansion (33) holds. From (33) and the bound H…”

Section: 2mentioning

confidence: 99%

Analysis of random walks in dynamic random environments via L2-perturbations

Avena

Blondel

Faggionato

2018

Stochastic Processes and their Applications

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We consider random walks in dynamic random environments given by Markovian dynamics on Z d . We assume that the environment has a stationary distribution µ and satisfies the Poincaré inequality w.r.t. µ. The random walk is a perturbation of another random walk (called "unperturbed"). We assume that also the environment viewed from the unperturbed random walk has stationary distribution µ. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of L 2 -bounded perturbations of Markov processes by means of the so-called Dyson-Phillips expansion.

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“…On another hand, the quenched CLT (a strengthening of the CLT), the quenched invariance principle and the law of the iterated logarithm (LIL) have been obtained, under various strengthening of (1), by Derriennic and Lin [24], Rassoul-Agha and Seppäläinen [50], Zhao and Woodroofe [62], Wu and Woodroofe [60], Cuny and Lin [12] and Cuny [9].…”

Section: Introductionmentioning

confidence: 99%

Invariance principles under the Maxwell–Woodroofe condition in Banach spaces

Cuny¹

2017

Ann. Probab.

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We prove that, for (adapted) stationary processes, the so-called Maxwell-Woodroofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. That result actually holds in the context of Banach valued stationary processes, including the case of L p -valued random variables, with 1 ≤ p < ∞. In this setting we also prove the weak invariance principle, hence generalizing a result of Peligrad and Utev [45]. The proofs make use of a new maximal inequality and of approximation by martingales, for which some of our results are also new.MSC 2010 subject classification: 60F17, 60F25, 60B12; Secondary: 37A50 Key words and phrases. Banach valued processes, compact law of the iterated logarithm, invariance principles, Maxwell-Woodroofe's condition.I am thankful to Jérôme Dedecker for providing him with a copy of [58]. I wish to thank two anonymous referees for valuable remarks that yielded an improved presentation of the paper.

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An almost sure invariance principle for additive functionals of Markov chains

Abstract: We prove an invariance principle for a vector-valued additive functional of a Markov chain for almost every starting point with respect to an ergodic equilibrium distribution. The hypothesis is a moment bound on the resolvent.

Cited by 11 publications

References 6 publications

Random walks in dynamic random environments and ancestry under local population regulation

Random walks in dynamic random environments and ancestry under local population regulation

Analysis of random walks in dynamic random environments via L2-perturbations

Invariance principles under the Maxwell–Woodroofe condition in Banach spaces

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