EMPIRICAL PROCESSES FOR RECURRENT AND TRANSIENT RANDOM WALKS IN RANDOM SCENERY

In this paper, we are interested in the asymptotic behaviour of the sequence of processes (W-n(s,t))(s,t is an element of[0,1]) with W-n(s,t) := Sigma(left perpendicularn tright perpendicular)(k=1) (1({xi Sk <= s}) - s) where (xi(x), x is an element of Z(d)) is a sequence of independent random variables uniformly distributed on [0, 1] and (S-n)(n is an element of N) is a random walk evolving in Z(d), independent of the xi's. In M. Wendler [Stoch. Process. Appl. 126 (2016) 2787-2799], the case where (S-n)(n is an element of N) is a recurrent random walk in Z such that (n(-1/alpha)S(n))(n >= 1) converges in distribution to a stable distribution of index alpha, with alpha is an element of (1, 2], has been investigated. Here, we consider the cases where (S-n)(n is an element of N) is either: (a) a transient random walk in Z(d), (b) a recurrent random walk in Z(d) such that (n(-1/d)S(n))(n >= 1) converges in distribution to a stable distribution of index d is an element of{1, 2}.