Convergence in law of the maximum of nonlattice branching random walk

Let eta(n)* denote the maximum, at time n, of a nonlattice one-dimensional branching random walk tin possessing (enough) exponential moments. In a seminal paper, Aidekon (Ann. Probab. 41 (2013) 1362-1426) demonstrated convergence of eta(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.