The Seneta–Heyde scaling for supercritical super-Brownian motion

Abstract

We consider the additive martingale ${\mathit{W}}_{\mathit{t}}(\mathit{\lambda })$ and the derivative martingale $\partial {\mathit{W}}_{\mathit{t}}(\mathit{\lambda })$ for one-dimensional supercritical super-Brownian motions with general branching mechanism. In the critical case $\mathit{\lambda }={\mathit{\lambda }}_0$, we prove that $\sqrt{\mathit{t}}{\mathit{W}}_{\mathit{t}}({\mathit{\lambda }}_0)$ converges in probability to a positive limit, which is a constant multiple of the almost sure limit $\partial {\mathit{W}}_{\infty }({\mathit{\lambda }}_0)$ of the derivative martingale $\partial {\mathit{W}}_{\mathit{t}}({\mathit{\lambda }}_0)$. We also prove that, on the survival event, ${lim\hspace{0.17em}sup}_{\mathit{t}\to \infty }\sqrt{\mathit{t}}{\mathit{W}}_{\mathit{t}}({\mathit{\lambda }}_0)=\infty$ almost surely.

Nous considérons la martingale additive ${\mathit{W}}_{\mathit{t}}(\mathit{\lambda })$ et la martingale dérivée $\partial {\mathit{W}}_{\mathit{t}}(\mathit{\lambda })$ pour les super-mouvements browniens surcritiques unidimensionnels avec mécanisme général de branchement. Dans le cas critique où $\mathit{\lambda }={\mathit{\lambda }}_0$, nous prouvons que $\sqrt{\mathit{t}}{\mathit{W}}_{\mathit{t}}({\mathit{\lambda }}_0)$ converge en probabilité vers une limite positive, qui est un multiple constant de la limite presque sûre $\partial {\mathit{W}}_{\infty }({\mathit{\lambda }}_0)$ de la martingale dérivée $\partial {\mathit{W}}_{\mathit{t}}({\mathit{\lambda }}_0)$. Nous prouvons également que, dans l’événement de survie, ${lim\hspace{0.17em}sup}_{\mathit{t}\to \infty }\sqrt{\mathit{t}}{\mathit{W}}_{\mathit{t}}({\mathit{\lambda }}_0)=\infty$ presque sûrement.