A LOCAL LIMIT THEOREM FOR RANDOM WALKS IN RANDOM SCENERY AND ON RANDOMLY ORIENTED LATTICES

• 成果类型: Article
• 署名作者: Castell, Fabienne; Guillotin-Plantard, Nadine; Pene, Francoise; Schapira, Bruno
• 署名单位: Aix-Marseille Universite; Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; CNRS - National Institute for Mathematical Sciences (INSMI); Centre National de la Recherche Scientifique (CNRS); Universite de Bretagne Occidentale; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP606
• 发表日期: 2011
• 页码: 2079-2118
• 关键词: dimensional random-walks; stable random-walks; shear-flow drift; large deviations; times; DIFFUSIONS; Invariance; moderate; sums

摘要

Random walks in random scenery are processes defined by Z(n) := Sigma(n)(k) = 1 X-xi(1)+... + X-k, where (X-k, k >= 1) and (xi(y), y is an element of 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 is an element of (0, 2] and beta is an element of (0, 2], respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when alpha not equal 1 and as n -> 8, of n (-delta) Z(n), for some suitable delta > 0 depending on alpha and beta. Here, we are interested in the convergence, as n -> 8, of n(delta)P (Z(n) = inverted right perpendicularn(delta)xinverted left perpendicular), when x is an element of R is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.