Fluctuations of additive martingale limits of branching Brownian motion

Abstract

Consider a one-dimensional branching Brownian motion. Let $W_{\infty }(\beta )$ denote the limit of the additive martingale in the subcritical regime $|\beta |<{\beta }_c$ and $Z_{\infty }$ be the limit of the derivative martingale at criticality. Madaule (Stochastic Process. Appl. 126 (2016), no. 2, 470--502) established the following convergence \[ \frac{W_\infty(β)}{β_c-β}\xrightarrow[β\nearrow β_c]{\mathbb{P}} 2Z_\infty. \] The goal of this paper is twofold: firstly, we strengthen this result into an almost sure convergence; secondly, we describe the fluctuations occurring in this convergence by proving \[ \frac{1}{β_c-β}\left( \frac{W_\infty(β)}{β_c-β} - 2 Z_\infty +2(β_c-β)\log(β_c-β) Z_\infty\right) \xrightarrow[β\nearrow β_c]{(d)} S, \] where, conditionally on $Z_{\infty }$, $S$ follows a spectrally negative 1-stable distribution with scale and shift parameters proportional to $Z_{\infty }$. Furthermore, these results are extended to the setting of complex additive martingales and the fluctuations to a multi-dimensional convergence.

23 pages, 3 figures