The Annals of Probability

Tightness for a family of recursion equations

Maury Bramson and Ofer Zeitouni

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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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Ann. Probab., Volume 37, Number 2 (2009), 615-653.

First available in Project Euclid: 30 April 2009

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Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G50: Sums of independent random variables; random walks 39B12: Iteration theory, iterative and composite equations [See also 26A18, 30D05, 37-XX]

Tightness recursion equations branching random walk cover time


Bramson, Maury; Zeitouni, Ofer. Tightness for a family of recursion equations. Ann. Probab. 37 (2009), no. 2, 615--653. doi:10.1214/08-AOP414.

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