Gail Ivanoff - Set-indexed Martingales

We give an appropriate framework and definitions for the theory of set-indexed martingales. Compensators and quadratic variation processes will be defined, and used to develop set-indexed analogues of the classical martingale characterizations of the Poisson process and Brownian motion. Sufficient conditions will be given in terms of compensators for a Poisson convergence theorem and in terms of quadratic variation processes for a martingale central limit theorem. Applications to set-indexed empirical processes will be discussed.