On the derivative martingale in a branching random walk

Abstract

We work under the Aïdékon–Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by Z. It is shown that $\mathbb{E}\mathit{Z}{\mathbb{1}}_{\{\mathit{Z}\le \mathit{x}\}}=log\mathit{x}+\mathit{o}(log\mathit{x})$ as $\mathit{x}\to \infty$. Also, we provide necessary and sufficient conditions under which $\mathbb{E}\mathit{Z}{\mathbb{1}}_{\{\mathit{Z}\le \mathit{x}\}}=log\mathit{x}+\mathrm{const}+\mathit{o}(1)$ as $\mathit{x}\to \infty$. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the derivative martingale to its limit Z. The methodological novelty of the present paper is a three terms representation of a subharmonic function of, at most, linear growth for a killed centered random walk of finite variance. This yields the aforementioned asymptotics and should also be applicable to other models.