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November 2017 Central limit theorem for random walks in doubly stochastic random environment: ${\mathscr{H}_{-1}}$ suffices
Gady Kozma, Bálint Tóth
Ann. Probab. 45(6B): 4307-4347 (November 2017). DOI: 10.1214/16-AOP1166

Abstract

We prove a central limit theorem under diffusive scaling for the displacement of a random walk on $\mathbb{Z}^{d}$ in stationary and ergodic doubly stochastic random environment, under the ${\mathscr{H}_{-1}}$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer], where it is assumed that the stream tensor is in $\mathscr{L}^{\max\{2+\delta,d\}}$, with $\delta>0$. Our proof relies on an extension of the relaxed sector condition of [Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012) 463–476], and is technically rather simpler than existing earlier proofs of similar results by Oelschläger [Ann. Probab. 16 (1988) 1084–1126] and Komorowski, Landim and Olla [Fluctuations in Markov Processes—Time Symmetry and Martingale Approximation (2012) Springer].

Citation

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Gady Kozma. Bálint Tóth. "Central limit theorem for random walks in doubly stochastic random environment: ${\mathscr{H}_{-1}}$ suffices." Ann. Probab. 45 (6B) 4307 - 4347, November 2017. https://doi.org/10.1214/16-AOP1166

Information

Received: 1 November 2014; Revised: 1 October 2016; Published: November 2017
First available in Project Euclid: 12 December 2017

zbMATH: 06838121
MathSciNet: MR3737912
Digital Object Identifier: 10.1214/16-AOP1166

Subjects:
Primary: 60F05, 60G99, 60K37

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 6B • November 2017
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