Renewal theorems for random walk in random scenery

Nadine Guillotin-Plantard,Françoise Pène

doi:10.1214/ejp.v17-1843

Nadine Guillotin-Plantard, Françoise Pène

Open Access

https://doi.org/10.1214/ejp.v17-1843

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Abstract

In this work, we establish renewal-type theorems, with precise asymptotics, in in the context of random walk in random sceneries.

Highlights

Renewal theorems in probability theory deal with the asymptotic behaviour when |a| → +∞ of the potential kernel formally defined as ∞Ka(h) := E[h(Zn − a)]n=1 where h is some complex-valued function defined on R and (Zn)n≥1 a real transient random process
N=1 where h is some complex-valued function defined on R and (Zn)n≥1 a real transient random process
In the classical case when Zn is the sum of n non-centered independent and identically distributed real random variables, renewal theorems were proved

Summary

Introduction

Renewal theorems in probability theory deal with the asymptotic behaviour when |a| → +∞ of the potential kernel formally defined as. N=1 where h is some complex-valued function defined on R and (Zn)n≥1 a real transient random process. In the classical case when Zn is the sum of n non-centered independent and identically distributed real random variables, renewal theorems were proved. In [8], Deligiannidis and Utev considered the case when α = 1 and β = 2 and proved the convergence in distribution of ((Znt/ n log(n))t≥0)n to a Brownian motion This result is obtained by an adaptation of the proof of the same result by Bothausen in [3] in the case when β = 2 and for a square integrable two-dimensional random walk (Sn)n. Lattice case: The random variables (ξx)x∈Z are assumed to be Z-valued and non-arithmetic i.e.

Preliminaries to the proofs

Study of the series ψ