The Seneta-Heyde scaling for supercritical super-Brownian motion

Abstract

We consider the additive martingale $W_t(\lambda )$ and the derivative martingale $\partial W_t(\lambda )$ for one-dimensional supercritical super-Brownian motions with general branching mechanism. In the critical case $\lambda ={\lambda }_0$, we prove that $\sqrt{t}{W}_t({\lambda }_0)$ converges in probability to a positive limit, which is a constant multiple of the almost sure limit $\partial W_{\infty }({\lambda }_0)$ of the derivative martingale $\partial W_t({\lambda }_0)$. We also prove that, on the survival event, ${\mathrm{limsup}}_{t\to \infty }\sqrt{t}{W}_t({\lambda }_0)=\infty$ almost surely.