Empirical processes for recurrent and transient random walks in random scenery^*

¹ Institut Camille Jordan, CNRS UMR 5208, Université de Lyon, Université Lyon 1, 43, Boulevard du 11 novembre 1918, 69622 Villeurbanne, France.
² Univ Brest, Université de Brest, IUF, LMBA, UMR CNRS 6205, 29238 Brest cedex, France.
³ Institut für mathematische Stochastik, Otto-von-Guericke-Universität, 39106 Magdeburg, Germany.

^** Corresponding author: guilloti@univ-lyon1.fr

Received: 27 November 2017
Accepted: 16 December 2019

Abstract

In this paper, we are interested in the asymptotic behaviour of the sequence of processes (W_n(s,t))_s,t∈[0,1] with

where (ξ_x, x ∈ ℤ^d) is a sequence of independent random variables uniformly distributed on [0, 1] and (S_n)_{n ∈ ℕ} is a random walk evolving in ℤ^d, independent of the ξ’s. In M. Wendler [Stoch. Process. Appl. 126 (2016) 2787–2799], the case where (S_n)_{n ∈ ℕ} is a recurrent random walk in ℤ such that (n^−1/αS_n)_n≥1 converges in distribution to a stable distribution of index α, with α ∈ (1, 2], has been investigated. Here, we consider the cases where (S_n)_{n ∈ ℕ} is either:

(a) a transient random walk in ℤ^d,

(b) a recurrent random walk in ℤ^d such that (n^−1/dS_n)_n≥1 converges in distribution to a stable distribution of index d ∈{1, 2}.