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2021 Gaussian fluctuations and a law of the iterated logarithm for Nerman’s martingale in the supercritical general branching process
Alexander Iksanov, Konrad Kolesko, Matthias Meiners
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Electron. J. Probab. 26: 1-22 (2021). DOI: 10.1214/21-EJP727


In his, by now, classical work from 1981, Nerman made extensive use of a crucial martingale (Wt)t0 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 W0L2, we prove a functional central limit theorem for (WWt+s)sR, 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 WWt.

Funding Statement

A. Iksanov was partially supported by the grant ME3625/3-1 and supported by Ulam programme funded by the Polish national agency for academic exchange (NAWA), project no. PPN/ULM/2019/1/00010/DEC/1. K. Kolesko was partially supported by the National Science Center, Poland (Sonata Bis, grant number DEC-2014/14/E/ST1/00588).


The authors thank the Associate Editor and the anonymous referee for several useful comments. This work was initiated during the visit of A. Iksanov to Innsbruck in August 2019. He gratefully acknowledges hospitality and the financial support.


Download Citation

Alexander Iksanov. Konrad Kolesko. Matthias Meiners. "Gaussian fluctuations and a law of the iterated logarithm for Nerman’s martingale in the supercritical general branching process." Electron. J. Probab. 26 1 - 22, 2021.


Received: 1 July 2021; Accepted: 26 November 2021; Published: 2021
First available in Project Euclid: 27 December 2021

Digital Object Identifier: 10.1214/21-EJP727

Primary: 60F05 , 60F17 , 60J80

Keywords: asymptotic fluctuations , functional central limit theorem , Law of the iterated logarithm , Nerman’s martingale , supercritical general branching process

Vol.26 • 2021
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