Introduction

Abstract

This part is concerned with convergence of a particular type of random map: the empirical process. The empirical measure ℙ _n of a sample of random elements X _1i,...,X _n on a measurable space (X, A) is the discrete random measure given by ℙ _n (C) = n ⁻¹#(1 ≤ i ≤ n: X _i ∈ C). Alternatively (if points are measurable), it can be described as the random measure that puts mass 1/n at each observation. We shall frequently write the empirical measure as the linear combination ${{\rm P}_n} = {n^{ - 1}}\sum _{i = 1}^n{\delta _{{X_i}}}$ of the dirac measures at the observations.