Let η ∗ n denote the maximum, at time n, of a nonlattice one-dimensional branching random walk ηn possessing (enough) exponential moments. In a seminal paper, Aïdekon (Ann. Probab. 41 (2013) 1362-1426) demonstrated convergence of η ∗ n in law, after centering, 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 (Convergence in law of the maximum of the two-dimensional discrete Gaussian free field; preprint). 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. We indicate the modifications needed in order to handle lattice random walks.
|Original language||English (US)|
|Number of pages||28|
|Journal||Annales de l'institut Henri Poincare (B) Probability and Statistics|
|State||Published - Nov 2016|
Bibliographical noteFunding Information:
Partially supported by NSF grants DMS-1105668 and DMS-1203201. Partially supported by NSF grant DMS-1313596. Partially supported by NSF grant DMS-1106627, a grant from the Israel Science Foundation, and the Herman P. Taubman chair of Mathematics at the Weizmann Institute.
- Branching random walks. Maximal displacement