Limit theorems for reinforced random walks on trees
Abstract
Reinforced random walk (RRW) is a broad class of processes which jump between nearest neighbor vertices of graphs, and prefer visiting often visited ones over seldom visited ones. This “nostalgia” makes a RRW strongly dependent on the whole past, so it is not a Markov process. There are two main classes of RRW, edge-reinforced random walks (ERRW) and vertex-reinforced random walks (VRRW). We study a continuous time version of VRRW, called a vertex-reinforced jump process (VRJP). We also study the ERRWs, and in particular Diaconis walk. The main purpose of this work is to describe how RRWs on trees drift away from the root. Let :.: be the distance of a vertex from the root. We prove the Strong Law of Large Numbers for :Xt: for vertex-reinforced jump processes on b-ary trees, with b ≥ 40, and Diaconis walk, for b ≥ 100. Our approach is unified. For VRJP and Diaconis walk on b-ary trees all that was previously known was transience when b ≥ 4 for the VRJP case, and b ≥ 2 for the Diaconis case (Davis and Volkov (2004) and Pemantle (1988), respectively). For once reinforced random walk, recurrence, strong law and central limit theorems were proved by Durrett, Kesten and Limic (2002).
Degree
Ph.D.
Advisors
Davis, Purdue University.
Subject Area
Statistics|Mathematics
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