Asymptotic mixed normality and hellinger processes

Abstract

We characterize the stable convergence of a sequence of density processes corresponding to binary filtered experiments to the exponential of a mixture of Gaussian processes, in terms of the convergence of their respective Hellinger processes, allowing random Hellinger processes in the limit. Within the applications we obtain necessary and sufficient conditions for the log-likelihood function of a supercritical branching process to be asymptotically mixed normal, in terms of the offspring distributions.