TIGHTNESS FOR A FAMILY OF RECURSION EQUATIONS

By Maury Bramson, Ofer Zeitouni. 2006

79 citations recorded by the source corpus.

In this paper, we study the tightness of solutions for a fam- ily of recursive equations. These equations arise naturally in the study of random walks on tree-like structures. Examples include the maxi- mal displacement of branching random walk in one dimension, and the cover time of 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 corre- sponding sequences of distribution functions after appropriate centering. We phrase our results in a fairly general context, which we hope will fa- cilitate their application in other settings.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d8924f0baea5314bc56c24e581698f0b5deabdf8

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Citations in the corpus

Maximum of the Characteristic Polynomial of Random Unitary Matrices (2015, unknown): “This started with the seminal work of Bramson [15] for branching Brownian motion and was later extended to general branching random walks [2,3,7,17,18].”
Minima in branching random walks (2007, background): “n, showing that the aforementioned family is tight when X has logarithmically concave distribution function. (We have not stated all the conditions on X and B required for his result.) More recently, Bramson and Zeitouni (2006, 2007) also proved tightness of the family {Mn −mn : n ∈ N}, under a stro”
Convergence in law of the minimum of a branching random walk (2013, unknown): “[10] Bramson, M.”
Convergence in law of the minimum of a branching random walk (2011, methodology): “Through recursive equations, Bramson and Zeitouni [11] obtained the tightness of Mn around its median, when assuming some properties on the decay of the tail distribution.”
Asymptotics of cover times via Gaussian free fields: Bounded-degree graphs and general trees (2011, background): “There are additional high precision estimates for cover times and Gaussian free ﬁelds on trees and 2D lattice: For regular binary trees T n of height n , Bramson and Zeitouni [12] proved that τ cov ( T n ) / 2 n is tight after proper centering.”
Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees (2007, background): “See, for example, Bachmann [4] and Bramson and Zeitouni [14], as well as Section 5 of the survey paper by Aldous and Bandyopadhyay [2].”
Convergence in Law for the Branching Random Walk Seen from Its Tip (2011, methodology): “Bramson,M., Zeitouni, O.: Tightness for a family of recursion equations.”
Branching random walks (2015, background): “12 without getting the optimal value 3/2, and tightness in Bramson and Zeitouni [71]; also, convergence in a special case (including the example of Gaussian displacements), centered at the mean, is proved by Bachmann [33].”