Fluctuations of the additive martingales related to super-Brownian motion

Let (Wₓ () ) ₓ ₀, parametrized by, be the additive martingale related to a supercritical super-Brownian motion on the real line and let W_ () 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. Indeed, assuming certain moment conditions on the branching mechanism, we show that the tail martingale W_ () -Wₓ (), properly normalized, converges in distribution to a non-degenerate random variable, and we identify the limit laws. We find that, for parameters with small absolute value, the fluctuations are affected by the behaviour of the branching mechanism around 0. In fact, we prove that, in the case of small ||, when is secondly differentiable at 0, the limit laws are scale mixtures of the standard normal laws, and when is `stable-like' near 0 in some proper sense, the limit laws are scale mixtures of the stable laws. However, the effect of the branching mechanism is limited in the case of large ||. In the latter case, we show that the fluctuations and limit laws are determined by the limiting extremal process of the super-Brownian motion.