Gaussian fluctuations and a law of the iterated logarithm for Nerman’s martingale in the supercritical general branching process

Abstract

In his, by now, classical work from 1981, Nerman made extensive use of a crucial martingale (Wt)t≥0 to prove convergence in probability, in mean and almost surely, of supercritical general branching processes (also known as Crump-Mode-Jagers branching processes) counted with a general characteristic. The martingale terminal value W figures in the limits of his results. We investigate the rate at which the martingale, now called Nerman’s martingale, converges to its limit W. More precisely, assuming the existence of a Malthusian parameter α>0 and W0∈L2, we prove a functional central limit theorem for (W−Wt+s)s∈R, properly normalized, as t→∞. The weak limit is a randomly scaled time-changed Brownian motion. Under an additional technical assumption, we prove a law of the iterated logarithm for W−Wt.