Convergence in law of the maximum of nonlattice branching random walk

By Maury Bramson, Jian Ding, Ofer Zeitouni. 2014

63 citations recorded by the source corpus.

Let $\eta^*_n$ denote the maximum, at time $n$, of a nonlattice one-dimensional branching random walk $\eta_n$ possessing (enough) exponential moments. In a seminal paper, Aidekon demonstrated convergence of $\eta^*_n$ in law, after recentering, and gave a representation of the limit. We give here a shorter proof of this convergence by employing reasoning motivated by Bramson, Ding and Zeitouni. Instead of spine methods and a careful analysis of the renewal measure for killed random walks, our approach employs a modified version of the second moment method that may be of independent interest.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e001cafd8fa5cb264960840230ca9b2698248231

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

The maximum of the CUE field (2016, methodology): “The approach we take to estimating the maximum of U is an adaptation of one developed to estimate the maximum of branching Brownian motion [Bra78] (see also [ABR09], [Aı̈d13] and [BDZ14] for the more closely related case of branching random walk).”
Convergence in Law of the Maximum of the Two‐Dimensional Discrete Gaussian Free Field (2013, unknown): “These tail estimates have since been adapted to the general BRW setting in [10].”
Convergence of the centered maximum of log-correlated Gaussian fields (2015, unknown): “Convergence for the centered maximum of BRW with Gaussian increments was obtained in [2], while the analogous result for general BRWs under mild assumptions was only obtained quite recently in the important work [1], using the notion of derivative martingale to describe the limit; see also [8].”
Branching random walks (2015, result): “This simplification is also found in the recent work of Bramson, Ding and Zeitouni [70], where a more compact proof is presented, with a particularly interesting new ingredient: instead of counting the number of certain vertices at generation n as we do in (5.”
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].”
Maxima of a randomized Riemann zeta function, and branching random walks (2015, background): “It has since been extended to more general branching random walks by several authors, for example, [1, 2, 11], and to log-correlated Gaussian fields see, for example, [12, 23].”