HAL : hal-00433533, version 2
 arXiv : 0911.3886
 Versions disponibles : v1 (19-11-2009) v2 (11-04-2012)
 Random walks with occasionally modified transition probabilities
 (19/11/2009)
 We study recurrence properties and the validity of the (weak) law of large numbers for (discrete time) processes which, in the simplest case, are obtained from simple symmetric random walk on $\Z$ by modifying the distribution of a step from a fresh point. If the process is denoted as $\{S_n\}_{n \ge 0}$, then the conditional distribution of $S_{n+1} - S_n$ given the past through time $n$ is the distribution of a simple random walk step, provided $S_n$ is at a point which has been visited already at least once during $[0,n-1]$. Thus in this case $P\{S_{n+1}-S_n = \pm 1|S_\ell, \ell \le n\} = 1/2$. We denote this distribution by $P_1$. However, if $S_n$ is at a point which has not been visited before time $n$, then we take for the conditional distribution of $S_{n+1}-S_n$, given the past, some other distribution $P_2$. We want to decide in specific cases whether $S_n$ returns infinitely often to the origin and whether $(1/n)S_n \to 0$ in probability. Generalizations or variants of the $P_i$ and the rules for switching between the $P_i$ are also considered.
 1 : Modélisation aléatoire de Paris X (MODAL'X) Université Paris X - Paris Ouest Nanterre La Défense 2 : Laboratoire de Mathématiques d'Orsay (LM-Orsay) CNRS : UMR8628 – Université Paris XI - Paris Sud
 Domaine : Mathématiques/Probabilités
 Mots Clés : Self-interacting random walk – Excited random walk – Weak law of large numbers – Cauchy law
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 hal-00433533, version 2 http://hal.archives-ouvertes.fr/hal-00433533 oai:hal.archives-ouvertes.fr:hal-00433533 Contributeur : Bruno Schapira <> Soumis le : Mercredi 11 Avril 2012, 11:09:29 Dernière modification le : Mercredi 11 Avril 2012, 12:28:56