Abstract
A functional approach for the study of the random walks in random sceneries (RWRS) is proposed. Under fairly general assumptions on the random walk and on the random scenery, functional limit theorems are proved. The method allows to study separately the convergence of the walk and of the scenery: on the one hand, a general criterion for the convergence of the local time of the walk is provided, on the other hand, the convergence of the random measures associated with the scenery is studied. This functional approach is robust enough to recover many of the known results on RWRS as well as new ones, including the case of many walkers evolving in the same scenery.
Highlights
Let S = (Sn)n∈N be a random walk in Zd starting from zero
Motivated by the construction of a new class of self-similar stationary increments processes, Kesten and Spitzer [17] and Borodin [3; 4] introduced random walks in random sceneries (RWRS) in dimension one and proved functional limit theorems. They considered the case when the random walk has i.i.d. increments belonging to the domain of attraction of a stable distribution with index α ∈
Condition (RW2.a): Kesten and Spitzer proved in [17] ((2.10) in Lemma 1) that if τx denotes the hitting time of the point x by the random walk S, the following inequality holds for any x ∈ Z, r ≥ 0 and n ≥ 1, P(N (n, x) ≥ r) ≤ P(N (n + 1, 0) ≥ r)P(τx ≤ n + 1). It implies that the moment of order 2 of the random variable N (n, x) is uniformly bounded by the moment of order 2 of N (n + 1, 0) which is equivalent to C n2(1−1/α) for some C > 0 (see (2.12) in Lemma 1 of [17])
Summary
Motivated by the construction of a new class of self-similar stationary increments processes, Kesten and Spitzer [17] and Borodin [3; 4] introduced RWRS in dimension one and proved functional limit theorems. They considered the case when the random walk has i.i.d. increments belonging to the domain of attraction of a stable distribution with index α ∈
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