Polotskiy S. Intrinsic martingales in the branching random walk

The uniformly integrable intrinsic martingales in the supercritical branching random walks are investigated. Conditions which ensure an exponential rate of convergence, of the intrinsic martingale to its limit are presented. Necessary and sufficient conditions for existence of certain moments of perpetuities are established. We find necessary and sufficient conditions for the existence of certain moments of a limiting random variable for the intrinsic martingale. Also we obtain sufficient conditions which guarantee the absolute continuity of the law of a limiting random variable for the intrinsic martingale.