These notes grew from lectures I gave at the University of Iowa in July of 1988, as part of the NSF-CBMS Regional Conference Series. The conference was ably organized by Tim Robertson and Richard Dykstra. I am most grateful to them for giving me the opportunity to experiment on a live and receptive audience with material not entirely polished. I also appreciate the suggestions and comments of Richard Dudley. Much of the lecture material was repackaging of ideas originally due to him.
In reworking the lecture notes I have tried (not always successfully) to resist the urge to push the presentation to ever higher levels of generality. My aim has been to introduce just enough technique to handle typical nontrivial asymptotic problems in statistics and econometrics. Of course the four substantial examples that represent the applications part of the lectures do not exhaust the possible uses for the theory. I have chosen them because they cleanly illustrate specific aspects of the theory, and also because I admire the original papers.
To anyone who is acquainted with the empirical process literature these notes might appear misleadingly titled. Empirical process theory usually deals with sums of independent (identically distributed) random variables f(ξi(ω)), with f running over a class of functions F. However I have chosen to present results for sums of independent stochastic processes fi(ω,t) indexed by a set T. Such a setting ac- commodates not only the relatively straightforward generalization to nonidentically distributed {ξi}, but also such simple modifications as a rescaling of the summands by a factor that depends on i and ω. It has often irked me that the traditional nota- tion cannot handle summands such as f(ξi)/i, even though the basic probabilistic method is unaffected.