On McKean's martingale in the Bovier-Hartung extremal process
Abstract
It has been proved by Bovier & Hartung [Elect. J. Probab. 19 (2014)] that the maximum of a variable-speed branching Brownian motion (BBM) in the weak correlation regime converges to a randomly shifted Gumbel distribution. The random shift is given by the almost sure limit of McKean's martingale, and captures the early evolution of the system. In the Bovier-Hartung extremal process, McKean's martingale thus plays a role which parallels that of the derivative martingale in the classical BBM. In this note, we provide an alternative interpretation of McKean's martingale in terms of a law of large numbers for high-points of BBM, i.e. particles which lie at a macroscopic distance from the edge. At such scales, 'McKean-like martingales' are naturally expected to arise in all models belonging to the BBM-universality class.
Keywords
Cite
@article{arxiv.1712.04325,
title = {On McKean's martingale in the Bovier-Hartung extremal process},
author = {Constantin Glenz and Nicola Kistler and Marius A. Schmidt},
journal= {arXiv preprint arXiv:1712.04325},
year = {2017}
}
Comments
13 pages