Empirical process


In probability theory, an empirical process is a stochastic process that describes the proportion of objects in a system in a given state.
For a process in a discrete state space a population continuous time Markov chain or Markov population model is a process which counts the number of objects in a given state.
In mean field theory, limit theorems are considered and generalise the central limit theorem for empirical measures. Applications of the theory of empirical processes arise in non-parametric statistics.

Definition

For X1, X2,... Xn independent and identically-distributed random variables in R with common cumulative distribution function F, the empirical distribution function is defined by
where IC is the indicator function of the set C.
For every x, Fn is a sequence of random variables which converge to F almost surely by the strong law of large numbers. That is, Fn converges to F pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergence of Fn to F by the Glivenko–Cantelli theorem.
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) for fixed measurable set A. Similarly, for a fixed function f, converges in distribution to a normal random variable, provided that and exist.
Definition
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.

Example

As an example, consider empirical distribution functions. For real-valued iid random variables X1, X2,..., Xn they are given by
In this case, empirical processes are indexed by a class It has been shown that is a Donsker class, in particular,