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| HAL : hal-00455154, version 1 |
| arXiv : 1002.1878 |
| Fiche détaillée |  Récupérer au format |
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| A local limit theorem for random walks in random scenery and on randomly oriented lattices |
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| Fabienne Castell 1Nadine Guillotin-Plantard 2, 3 |
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| (09/02/2010) |
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| 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. |
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| 1 : | Laboratoire d'Analyse, Topologie, Probabilités (LATP) |
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CNRS : UMR6632 – Université de Provence - Aix-Marseille I – Université Paul Cézanne - Aix-Marseille III |
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| 2 : | UNIVERSITÉ CLAUDE BERNARD - Lyon 1 (UCB) |
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LaPCS-50 |
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| 3 : | Institut Camille Jordan (ICJ) |
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CNRS : UMR5208 – Université Claude Bernard - Lyon I – Ecole Centrale de Lyon – Institut National des Sciences Appliquées de Lyon |
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| 4 : | Laboratoire de mathématiques de Brest (LM) |
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CNRS : UMR6205 – Université de Bretagne Occidentale - Brest – Institut Supérieur des Sciences et Technologies de Brest (ISSTB) |
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| 5 : | Laboratoire de Mathématiques d'Orsay (LM-Orsay) |
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CNRS : UMR8628 – Université Paris XI - Paris Sud |
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| Domaine | : | Mathématiques/Probabilités |
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| Random walk in random scenery – random walk on randomly oriented lattices – local limit theorem – stable process |
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| Liste des fichiers attachés à ce document : | ||||||||||
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| hal-00455154, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00455154 | |
| oai:hal.archives-ouvertes.fr:hal-00455154 | |
| Contributeur : Bruno Schapira | |
| Soumis le : Mardi 9 Février 2010, 16:07:28 | |
| Dernière modification le : Mardi 9 Février 2010, 16:17:31 | |
