Fluctuations of the Additive Martingales Related to Super-Brownian Motion

Let (W_t(λ))_t≥0, parametrized by λ ∈ ℝ, be the additive martingale related to a supercritical super-Brownian motion on the real line and let W_∞(λ) be its limit. Under a natural condition for the martingale limit to be non-degenerate, we investigate the rate at which the martingale approaches its limit. Under certain moment conditions on the branching mechanism, we show that W_∞(X) − W_t(λ), properly normalized, converges in distribution to a non-degenerate random variable, and we identify the limit law. We find that, for parameters with small absolute value, the fluctuations are affected by the behaviour of the branching mechanism ψ around 0. In fact, we prove that, in the case of small ∣λ∣, when ψ is twice differentiable at 0, the limit law is a scale mixture of the standard normal law, and when ψ is ‘stable-like’ near 0 in some sense, the limit law is a scale mixture of certain stable law. However, the effect of the branching mechanism is not very strong in the case of large ∣λ∣. In the latter case, we show that the fluctuations and limit laws are determined by the limiting extremal process of the super-Brownian motion.