The Seneta-Heyde scaling for supercritical super-Brownian motion

2021/09/10 by Hou, Haojie, Ren, Yan-Xia, Song, Renming
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2109.04594

Abstract

We consider the additive martingale W_t(λ) and the derivative martingale ∂ W_t(λ) for one-dimensional supercritical super-Brownian motions with general branching mechanism. In the critical case λ=λ₀, we prove that √(t)W_t(λ₀) converges in probability to a positive limit, which is a constant multiple of the almost sure limit ∂ W_∞(λ₀) of the derivative martingale ∂ W_t(λ₀). We also prove that, on the survival event, \limsup_t→∞√(t)W_t(λ₀)=∞ almost surely.

Related

• Fluctuations of the additive martingales related to super-Brownian motion
• Small value probabilities of additive and derivative martingales in supercritical branching Brownian motions and super Brownian motions
• Limiting distributions for a class of super-Brownian motions with spatially dependent branching mechanisms
• The extremal process of super-Brownian motion
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