New Preprint: Empirical processes for recurrent and transient random walks in random scenery

A new preprint joint with Nadine Guillotin-Plantard and Françoise Péne about „Empirical processes for recurrent and transient random walks in random scenery” is online at arXiv.

Abstract: In this paper, we are interested in the asymptotic behaviour of the sequence of processes (W_n(s,t))_{s,t\in[0,1]} with

W_n(s,t):=\sum_{k=1}^{[nt]}(1_{\{\xi_{S_k}\leq s\}}-s)

where \xi_x, x\in\Z^d is a sequence of independent random variables uniformly distributed on [0,1] and S_n, n\in\N is a random walk evolving in \Z^d, independent of the\xi’s. In Wendler (2016), the case where S_n, n\in\N is a recurrent random walk in \Z such that n^{-\frac 1\alpha}S_n, n\geq 1 converges in distribution to a stable distribution of index \alpha, with \alpha\in(1,2], has been investigated. Here, we consider the cases where S_n, n\in\N is either:

a) a transient random walk in \Z^d,
b) a recurrent random walk in \Z^d such that n^{-\frac 1d}S_n, n\geq 1 converges in distribution to a stable distribution of index d\in\{1,2\}.