Definition
It is known that under certain conditions empirical measures uniformly converge to the probability measure P (see Glivenko–Cantelli theorem). The theory of empirical processes provides the rate of this convergence.
A centered and scaled version of the empirical measure is the signed measure
It induces a map on measurable functions f given by
By the central limit theorem, converges in distribution to a normal random variable N(0, P(A)(1 − P(A))) for fixed measurable set A. Similarly, for a fixed function f, converges in distribution to a normal random variable, provided that and exist.
Definition
- is called an empirical process indexed by, a collection of measurable subsets of S.
- is called an empirical process indexed by, a collection of measurable functions from S to .
A significant result in the area of empirical processes is Donsker's theorem. It has led to a study of Donsker classes: sets of functions with the useful property that empirical processes indexed by these classes converge weakly to a certain Gaussian process. While it can be shown that Donsker classes are Glivenko–Cantelli classes, the converse is not true in general.
Read more about this topic: Empirical Process
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