2020/05/11 by Iksanov, Alexander, Kolesko, Konrad, Meiners, Matthias · 1 citation
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2005.05119
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 (a.k.a. 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.