Empirical processes for recurrent and transient random walks in random scenery

2017/11/28 by Guillotin-Plantard, Nadine, Pene, Francoise, Wendler, Martin
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1711.10202

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 W_n(s,t):=∑_k=1^{\lfloor nt\rfloor}(1_\ξ_Sk≤ s\-s) where (ξ_x, x∈ℤ^d) is a sequence of independent random variables uniformly distributed on [0,1] and (S_n)_{n∈\mathbb N} is a random walk evolving in ℤ^d, independent of the ξ's. In Wendler (2016), the case where (S_n)_{n∈\mathbb N} is a recurrent random walk in ℤ such that (n^{-\frac 1α}S_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∈\mathbb 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≥ 1} converges in distribution to a stable distribution of index d∈\1,2\.

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