ABSTRACT
This chapter examines set-indexed empirical measures and function-indexed analogues which are more tractable in many ways. It utilizes the powerful concept of strong approximation for uniform empirical processes to produce strong approximations of Poisson processes by Wiener and Kiefer processes. An important tool for analysis of limit properties such as is strong approximation: a statement of convergence in distribution is replaced by a statement of almost sure convergence accompanied by a rate of convergence. Provided that the rate of convergence is sufficiently rapid, results on path properties of the Brownian bridge for example, concerning modulus of continuity and oscillation behavior carry over to the En. The chapter explores specialized empirical Laplace and zero-probability functionals for Poisson processes on general spaces. The quantile transformation is a standard and important tool for reducing assertions concerning general empirical processes to corresponding properties of uniform empirical processes.
