Convergence in law of the minimum of a branching random walk

By Elie Aidékon. 2011

266 citations recorded by the source corpus.

We consider the minimum of a super-critical branching random walk. Addario-Berry and Reed [Ann. Probab. 37 (2009) 1044–1079] proved the tightness of the minimum centered around its mean value. We show that a convergence in law holds, giving the analog of a well-known result of Bramson [Mem. Amer. Math. Soc. 44 (1983) iv+190] in the case of the branching Brownian motion.

Subfields: Management Science and Operations Research, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ef106fb99caa3890c98e183cf19e9e0b7cf2fdcd

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In Management Science and Operations Research

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Convergence in Law of the Maximum of the Two‐Dimensional Discrete Gaussian Free Field (2013, background): “While sharing some important components with both [8] and [3], the tail estimates we employ here differ substantially from those of both papers.”
Extrema of log-correlated random variables: Principles and Examples (2016, background): “result for branching Brownian motion (see Lalley & Sellke [47] for the expression in terms of the derivative martingale). For branching random walks, it was proved in great generality by Adekon [3], see also [1,14,24]; for the two-dimensional Gaussian free eld, by Bramson et al. [23] (see Biskup”
On the extreme values of the Riemann zeta function on random intervals of the critical line (2016, unknown): “In the case of branching Brownian motion and branching random walks, for which a tree structure is explicit, the study goes back to the seminal paper by Bramson [12], and more precise and general results have later been proven by Aı̈dékon, Hu and Shi (see [18, 2, 1]).”
Branching random walks (2015, methodology): “The statement of the peeling lemma, slightly more general than in Äıdékon [8] (where only the case K = 0 is studied because the absorbing barrier is placed in a different place), is borrowed from Chen [87].”
Maximum of a log-correlated Gaussian field (2013, methodology): “1366 in [1] we have also for any t ≥ 1 and ρ(x) ∈ [κd log l, log t],”
Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012, background): “One can thus conjecture by analogy with the branching random walk case [1]:”
Gaussian multiplicative chaos and applications: A review (2013, background): “By analogy with the branching random walk case ([4]), it is conjectured in [58] : Conjecture 6.”
The Seneta–Heyde scaling for the branching random walk (2011, background): “(The uniformity in k ∈ (n,2n]∩Z is not stated in [2], but the same proof holds.”
Convergence in Law for the Branching Random Walk Seen from Its Tip (2011, methodology): “A crucial observation, inspired from [2], is that the convergence in law of this k-dimensional random vector can be reduced to the study of its tail distribution.”
Maxima of a randomized Riemann zeta function, and branching random walks (2015, background): “[1, 2, 11], and to log-correlated Gaussian fields, see for example [10, 22].”
Branching Brownian motion seen from its tip (2011, background): “The same kind of argument with the same stopping lines appear in [25] and in [1].”
The many-to-few lemma and multiple spines (2011, background): “he original branching process, and this additional structure can be used to construct and understand changes of measure on the branching system, which turns out to be a powerful tool: see for example [2, 12]. The aim of this article is to state a quite general kth moment formula which we call the ma”
A simple path to asymptotics for the frontier of a branching Brownian motion (2011, background): “ds ≤ ∫ 1 0 P̂ (A1 ∩A2|τ = s)ds ≤ ∫ 1 0 P̂ [∫ ∞ 0 P̂ ( Y 1 t , Y 2 t ∈ [1, 2] ∣”
Convergence in law of the maximum of nonlattice branching random walk (2014, methodology): “Results comparable to those in [4] were obtained in the context of the two-dimensional discrete Gaussian free field in Bramson, Ding, and Zeitouni [5].”