We establish the moderate deviation principle for the solutions of a class of stochastic partial differential equations with non-Lipschitz continuous coefficients. As an application, we derive the moderate deviation principle for two important population models: super-Brownian motion and the Fleming–Viot process.

Let {Zn, n = 0, 1, 2, . . .} be a supercritical branching process, {Nt, t ≥ 0} be a Poisson process independent of {Zn, n = 0, 1, 2, . . .}, then {ZNt, t ≥ 0} is a supercritical Poisson random indexed branching process. We show a law of large numbers, central limit theorem, and large and moderate deviation principles for log ZNt.

The purpose of this paper is to investigate moderate deviations for the Durbin–Watsonstatistic associated with the stable first-order autoregressive process where the drivennoise is also given by a first-order autoregressive process. We first establish a moderatedeviation principle for both the least squares estimator of the unknown parameter of theautoregressive process as well as for the serial correlation estimator associated with thedriven noise. It enables us to provide a moderate deviation principle for theDurbin–Watson statistic in the case where the driven noise is normally distributed and inthe more general case where the driven noise satisfies a less restrictive Chen–Ledoux typecondition.