Abstract
In this paper, we are interested in the asymptotic behaviour of the sequence of processes (Wn(s,t))s,t∈[0,1] with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(\mathds{1}_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where (ξx, x ∈ ℤd) is a sequence of independent random variables uniformly distributed on [0, 1] and (Sn)n ∈ ℕ is a random walk evolving in ℤd, independent of the ξ’s. In M. Wendler [Stoch. Process. Appl. 126 (2016) 2787–2799], the case where (Sn)n ∈ ℕ is a recurrent random walk in ℤ such that (n−1/αSn)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 (Sn)n ∈ ℕ is either: (a) a transient random walk in ℤd, (b) a recurrent random walk in ℤd such that (n−1/dSn)n≥1 converges in distribution to a stable distribution of index d ∈{1, 2}.
Highlights
Introduction and main resultsThe sequential empirical process has been studied under various assumptions, starting with Muller [33] under independence
The processk≥1 can be viewed as the increments of a random walk in random scenery (RWRS, in short) (Zn)n≥1
If we consider a random walk in random scenery (XSk )k∈N with random variables (Xx)x∈Zd not uniformly distributed on the interval [0, 1], the limit distribution of the sequential empirical process can still be deduced from Theorem 1.1
Summary
The sequential empirical process has been studied under various assumptions, starting with Muller [33] under independence. If we consider a random walk in random scenery (XSk )k∈N with random variables (Xx)x∈Zd not uniformly distributed on the interval [0, 1], the limit distribution of the sequential empirical process can still be deduced from Theorem 1.1. 1{FX−1(ξSk )≤s} − FX (s) k=1 k=1 and P(FX−1(ξx) ≤ s) = P(ξx ≤ FX (s)) = FX (s) = P(X ≤ s), so the random variables (FX−1(ξx))x∈Zd are independent with distribution function FX. It suffices to study the case where the scenery is uniformly distributed on [0, 1]
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