Recurrence of multidimensional persistent random walks. Fourier and series criteria

Abstract

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.