Fluctuations of additive martingale limits of branching Brownian motion

Consider a one-dimensional branching Brownian motion. Let $W_{\mathrm{\infty }}(\beta )$ denote the limit of the additive martingale in the subcritical regime $\mid \beta \mid <{\beta }_c$ and $Z_{\mathrm{\infty }}$ be the limit of the derivative martingale at criticality. Madaule (Stochastic Process. Appl. 126 (2016), no. 2, 470–502) established the following convergence $$\frac{W_{\mathrm{\infty }}(\beta )}{{\beta }_c-\beta }\underset{\beta \nearrow {\beta }_c}{\overset{\mathbb{P}}{\to }}2Z_{\mathrm{\infty }}\mathrm{.}$$ 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 $$\frac{1}{{\beta }_c-\beta }(\frac{W_{\mathrm{\infty }}(\beta )}{{\beta }_c-\beta }-2Z_{\mathrm{\infty }}+2({\beta }_c-\beta )\log ({\beta }_c-\beta )Z_{\mathrm{\infty }})\underset{\beta \nearrow {\beta }_c}{\overset{(d)}{\to }}S,$$ where, conditionally on $Z_{\mathrm{\infty }}$, $S$ follows a spectrally negative 1-stable distribution with scale and shift parameters proportional to $Z_{\mathrm{\infty }}$. Furthermore, these results are extended to the setting of complex additive martingales and the fluctuations to a multi-dimensional convergence.