Preliminaries for Empirical Processes

In this chapter, we cover several mathematical topics that play a central role in the empirical process results we present later. Metric spaces are crucial since they provide the descriptive language by which the most important results about stochastic processes are derived and expressed. Outer expectations, or, more correctly, outer integrals are key to defining and utilizing outer modes of convergence for quantities which are not measurable. Since many statistical quantities of interest are not measurable with respect to the uniform topology, which is often the topology of choice for applications, outer modes of convergence will be the primary approach for stochastic convergence throughout this book. Linear operators and functional derivatives also play a major role in empirical process methods and are key tools for the functional delta method and Z-estimator theory discussed in Chapters 12 and 13.