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 ${(W_t)}_{t\ge 0}$ 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 $\mathit{\alpha }>0$ and $W_0\in L^2$, we prove a functional central limit theorem for ${(W-W_{t+s})}_{s\in \mathbb{R}}$, properly normalized, as $t\to \mathrm{\infty }$. 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 $W-W_t$.