Central limit theorems for set-indexed strong martingales.

In [24], Ivanoff and Merzbach introduced the notion of set-indexed strong martingales, a generalization of the planar strong martingales introduced by Cairoli and Walsh in [10]. A set-indexed strong martingale is a special case of a set-indexed process by which we mean a collection X=XA:A∈A of random variables where A is some collection of subsets of a fixed set T. In this thesis, T is always a compact metric space and A is a semilattice of closed subsets of T. In this thesis, we obtain limit theorems for sequences of set-indexed strong martingales and develop two general tools that are useful in obtaining such limit theorems. These limit theorems establish convergence to set-indexed Gaussian processes in one of two modes, functional or semifunctional. Whereas the former mode of convergence is classic, the latter is entirely new and is particularly well-suited to set-indexed strong martingale central limit theorems. The first tool developed is the establishment of sufficient conditions for compactness in the function space D( A ) defined in [22] under a Skorokhod J2-like metric. D( A ) serves as a set-indexed generalization of the classic function space D[0, 1]. The resulting compact sets lead to tightness criteria for set-indexed processes with sample paths in D( A ). These results extend those found in Section 3 of [5]. For the second tool, quadratic variation for set-indexed strong martingales is defined and conditions ensuring its existence are given. The general role of quadratic variation in set-indexed strong martingale central limit theorems is similar to that played by quadratic variation processes in the classical theory. Namely, under certain conditions, convergence of quadratic variations to a continuous deterministic limit implies convergence of the underlying sequence of set-indexed strong martingales to a suitably scaled set-indexed Gaussian process. As an application, we derive a central limit theorem for set-indexed weighted empirical processes