A branching random walk is a discrete-time particle system on the real line in which every particle gives independently birth to offspring positioned around the position of their parent. The derivative martingale of the branching random walk is a stochastic process which allows to measure the number of particles traveling at maximal speed in this process. Under some general integrability conditions uncovered by Aïdékon (2013) and Chen (2015), we know the derivative martingale converges a.s. to a non-negative limit.

Recently, Maillard and Pain (2019) obtained a necessary and sufficient condition on the reproduction law of the branching Brownian motion under which the convergence of the derivative martingale satisfies a specific central limit theorem. In this talk, based on a joint work with Alexander Iksanov and Dariusz Buraczewski, we extend their result to branching random walk settings. The proof is based on the characterization of subharmonic functions of the killed random walk with at most linear growth.