Fluctuations of the Additive Martingales Related to Super-Brownian Motion

Let (Wt(lambda))t >= 0, parametrized by lambda is an element of & Ropf;, be the additive martingale related to a supercritical super-Brownian motion on the real line and let W infinity(lambda) 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 infinity(X) - Wt(lambda), 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 psi around 0. In fact, we prove that, in the case of small divided by lambda divided by, when psi is twice differentiable at 0, the limit law is a scale mixture of the standard normal law, and when psi 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 divided by lambda divided by. In the latter case, we show that the fluctuations and limit laws are determined by the limiting extremal process of the super-Brownian motion.