Renewal theorems for random walks in random scenery

2011/12/03 by Nadine Guillotin‐Plantard, Guillotin-Plantard, Nadine, Françoise Pène +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · doi:10.48550/arxiv.1112.0658

openalex publication_date 2011/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Random walks in random scenery are processes defined by Z_n:=∑_k=1ⁿξ_X1+...+Xk, where (X_k,k≥ 1) and (ξ_y,y∈\mathbb Z) are two independent sequences of i.i.d. random variables. We suppose that the distributions of X₁ and ξ₀ belong to the normal domain of attraction of strictly stable distributions with index α∈[1,2] and β∈(0,2) respectively. We are interested in the asymptotic behaviour as |a| goes to infinity of quantities of the form ∑_{n≥ 1}\mathbb E[h(Z_n-a)] (when (Z_n)_n is transient) or ∑_{n≥ 1}\mathbb E[h(Z_n)-h(Z_n-a)] (when (Z_n)_n is recurrent) where h is some complex-valued function defined on ℝ or ℤ.

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