Limit theorems for one and two-dimensional random walks in random scenery

Fabienne Castell; Nadine Guillotin-Plantard; Françoise Pène

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 2, page 506-528
  • ISSN: 0246-0203

Abstract

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Random walks in random scenery are processes defined by Z n : = ∑ k = 1 n ξ X 1 + ⋯ + X k , where ( X k , k ≥ 1 ) and ( ξ y , y ∈ ℤ d ) are two independent sequences of i.i.d. random variables with values in ℤ d and ℝ respectively. We suppose that the distributions of X 1 and ξ 0 belong to the normal basin of attraction of stable distribution of index α ∈ ( 0 , 2 ] and β ∈ ( 0 , 2 ] . When d = 1 and α ≠ 1 , a functional limit theorem has been established in (Z. Wahrsch. Verw. Gebiete50 (1979) 5–25) and a local limit theorem in (Ann. Probab.To appear). In this paper, we establish the convergence in distribution and a local limit theorem when α = d (i.e. α = d = 1 or α = d = 2 ) and β ∈ ( 0 , 2 ] . Let us mention that functional limit theorems have been established in (Ann. Probab.17(1989) 108–115) and recently in (An asymptotic variance of the self-intersections of random walks. Preprint) in the particular case when β = 2 (respectively for α = d = 2 and α = d = 1 ).

How to cite

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Castell, Fabienne, Guillotin-Plantard, Nadine, and Pène, Françoise. "Limit theorems for one and two-dimensional random walks in random scenery." Annales de l'I.H.P. Probabilités et statistiques 49.2 (2013): 506-528. <http://eudml.org/doc/271976>.

@article{Castell2013,
abstract = {Random walks in random scenery are processes defined by $Z_\{n\}:=\sum _\{k=1\}^\{n\}\xi _\{X_\{1\}+\cdots +X_\{k\}\}$, where $(X_\{k\},k\ge 1)$ and $(\xi _\{y\},y\in \{\mathbb \{Z\}\}^\{d\})$ are two independent sequences of i.i.d. random variables with values in $\{\mathbb \{Z\}\}^\{d\}$ and $\mathbb \{R\}$ respectively. We suppose that the distributions of $X_\{1\}$ and $\xi _\{0\}$ belong to the normal basin of attraction of stable distribution of index $\alpha \in (0,2]$ and $\beta \in (0,2]$. When $d=1$ and $\alpha \ne 1$, a functional limit theorem has been established in (Z. Wahrsch. Verw. Gebiete50 (1979) 5–25) and a local limit theorem in (Ann. Probab.To appear). In this paper, we establish the convergence in distribution and a local limit theorem when $\alpha =d$ (i.e. $\alpha =d=1$ or $\alpha =d=2$) and $\beta \in (0,2]$. Let us mention that functional limit theorems have been established in (Ann. Probab.17(1989) 108–115) and recently in (An asymptotic variance of the self-intersections of random walks. Preprint) in the particular case when $\beta =2$ (respectively for $\alpha =d=2$ and $\alpha =d=1$).},
author = {Castell, Fabienne, Guillotin-Plantard, Nadine, Pène, Françoise},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random walk in random scenery; local limit theorem; local time; stable process},
language = {eng},
number = {2},
pages = {506-528},
publisher = {Gauthier-Villars},
title = {Limit theorems for one and two-dimensional random walks in random scenery},
url = {http://eudml.org/doc/271976},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Castell, Fabienne
AU - Guillotin-Plantard, Nadine
AU - Pène, Françoise
TI - Limit theorems for one and two-dimensional random walks in random scenery
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 2
SP - 506
EP - 528
AB - Random walks in random scenery are processes defined by $Z_{n}:=\sum _{k=1}^{n}\xi _{X_{1}+\cdots +X_{k}}$, where $(X_{k},k\ge 1)$ and $(\xi _{y},y\in {\mathbb {Z}}^{d})$ are two independent sequences of i.i.d. random variables with values in ${\mathbb {Z}}^{d}$ and $\mathbb {R}$ respectively. We suppose that the distributions of $X_{1}$ and $\xi _{0}$ belong to the normal basin of attraction of stable distribution of index $\alpha \in (0,2]$ and $\beta \in (0,2]$. When $d=1$ and $\alpha \ne 1$, a functional limit theorem has been established in (Z. Wahrsch. Verw. Gebiete50 (1979) 5–25) and a local limit theorem in (Ann. Probab.To appear). In this paper, we establish the convergence in distribution and a local limit theorem when $\alpha =d$ (i.e. $\alpha =d=1$ or $\alpha =d=2$) and $\beta \in (0,2]$. Let us mention that functional limit theorems have been established in (Ann. Probab.17(1989) 108–115) and recently in (An asymptotic variance of the self-intersections of random walks. Preprint) in the particular case when $\beta =2$ (respectively for $\alpha =d=2$ and $\alpha =d=1$).
LA - eng
KW - random walk in random scenery; local limit theorem; local time; stable process
UR - http://eudml.org/doc/271976
ER -

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