The Seneta-Heyde scaling for supercritical super-Brownian motion

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, $\limsup_{t\to\infty}\sqrt{t}W_t(\lambda_0)=\infty$ almost surely.