2004/11/16 by Remco van der Hofstad, van der Hofstad, Remco, Nina Gantert +3
Mathematics · Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #Stochastic processes and statistical mechanics #Diffusion and Search Dynamics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.math/0411361
Let (Zn)n\∈ N0 be a d-dimensional random walk in random scenery, i.e.,\nZn=\∑k=0n-1YSk with (Sk)k\∈ N0 a random walk in Zd and\n(Yz)z\∈ Zd an i.i.d. scenery, independent of the walk.\n We assume that the random variables Yz have a stretched exponential tail. In\nparticular, they do not possess exponential moments. We identify the speed and\nthe rate of the logarithmic decay of Pr(Zn>tn n) for all sequences\n(tn)n\∈ N satisfying a certain lower bound. This complements previous\nresults, where it was assumed that Yz has exponential moments of all orders.\nIn contrast to the previous situation,the event Zn>tnn is not realized by\na homogeneous behavior of the walk's local times and the scenery, but by many\nvisits of the walker to a particular site and a large value of the scenery at\nthat site. This reflects a well-known extreme behavior typical for random\nvariables having no exponential moments.\n