%0 Journal Article
%T A local limit theorem for random walks in random scenery and on randomly oriented lattices
%+ Laboratoire d'Analyse, Topologie, Probabilités (LATP)
%+ Université Claude Bernard Lyon 1 (UCBL)
%+ Institut Camille Jordan (ICJ)
%+ Laboratoire de mathématiques de Brest (LM)
%+ Laboratoire de Mathématiques d'Orsay (LM-Orsay)
%A Castell, Fabienne
%A Guillotin-Plantard, Nadine
%A Pene, Françoise
%A Schapira, Bruno
%< avec comité de lecture
%@ 0091-1798
%J Annals of Probability
%I Institute of Mathematical Statistics
%P Vol. 39, No 6, 2079--2118
%8 2011
%D 2011
%Z 1002.1878
%R 10.1214/10-AOP606
%K Random walk in random scenery
%K random walk on randomly oriented lattices
%K local limit theorem
%K stable process
%Z 60F05; 60G52
%Z Mathematics [math]/Probability [math.PR]Journal articles
%X Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index $\alpha\in (0,2]$ and $\beta\in (0,2]$ respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when $\alpha\neq 1$ and as $n\to \infty$, of $n^{-\delta}Z_n$, for some suitable $\delta>0$ depending on $\alpha$ and $\beta$. Here we are interested in the convergence, as $n\to \infty$, of $n^\delta{\mathbb P}(Z_n=\lfloor n^{\delta} x\rfloor)$, when $x\in \RR$ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.
%G English
%2 https://hal.science/hal-00455154/document
%2 https://hal.science/hal-00455154/file/TLL09_02final.pdf
%L hal-00455154
%U https://hal.science/hal-00455154
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%~ UNIV-BREST
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