Open Access
May 2020 Recurrence of multidimensional persistent random walks. Fourier and series criteria
Peggy Cénac, Basile de Loynes, Yoann Offret, Arnaud Rousselle
Bernoulli 26(2): 858-892 (May 2020). DOI: 10.3150/18-BEJ1098


The recurrence and transience of persistent random walks built from variable length Markov chains are investigated. It turns out that these stochastic processes can be seen as Lévy walks for which the persistence times depend on some internal Markov chain: they admit Markov random walk skeletons. A recurrence versus transience dichotomy is highlighted. Assuming the positive recurrence of the driving chain, a sufficient Fourier criterion for the recurrence, close to the usual Chung–Fuchs one, is given and a series criterion is derived. The key tool is the Nagaev–Guivarc’h method. Finally, we focus on particular two-dimensional persistent random walks, including directionally reinforced random walks, for which necessary and sufficient Fourier and series criteria are obtained. Inspired by (Adv. Math. 208 (2007) 680–698), we produce a genuine counterexample to the conjecture of (Adv. Math. 117 (1996) 239–252). As for the one-dimensional case studied in (J. Theoret. Probab. 31 (2018) 232–243), it is easier for a persistent random walk than its skeleton to be recurrent. However, such examples are much more difficult to exhibit in the higher dimensional context. These results are based on a surprisingly novel – to our knowledge – upper bound for the Lévy concentration function associated with symmetric distributions.


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Peggy Cénac. Basile de Loynes. Yoann Offret. Arnaud Rousselle. "Recurrence of multidimensional persistent random walks. Fourier and series criteria." Bernoulli 26 (2) 858 - 892, May 2020.


Received: 1 December 2017; Revised: 1 July 2018; Published: May 2020
First available in Project Euclid: 31 January 2020

zbMATH: 07166550
MathSciNet: MR4058354
Digital Object Identifier: 10.3150/18-BEJ1098

Keywords: concentration functions , Fourier and series recurrence criteria , Fourier perturbations , Markov operators , Markov random walks , persistent random walks , Variable length markov chain

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 2 • May 2020
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