Tightness for a family of recursion equations

Details Tightness for a family of recursion equations Tightness for a family of recursion equations

2006-12-14

arxiv.org/abs/math/0612382v2

Annals of Probability 2009, Vol. 37, No. 2, 615-653

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AOP414 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/08-AOP414

In this paper we study the tightness of solutions for a family of recursion equations. These equations arise naturally in the study of random walks on tree-like structures. Examples include the maximal displacement of a branching random walk in one dimension and the cover time of a symmetric simple random walk on regular binary trees. Recursion equations associated with the distribution functions of these quantities have been used to establish weak laws of large numbers. Here, we use these recursion equations to establish the tightness of the corresponding sequences of distribution functions after appropriate centering. We phrase our results in a fairly general context, which we hope will facilitate their application in other settings.

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