In mean field theory, limit theorems (as the number of objects becomes large) are considered and generalise the central limit theorem for empirical measures.
Applications of the theory of empirical processes arise in non-parametric statistics.
[1] For X1, X2, ... Xn independent and identically-distributed random variables in R with common cumulative distribution function F(x), the empirical distribution function is defined by where IC is the indicator function of the set C. For every (fixed) x, Fn(x) is a sequence of random variables which converge to F(x) almost surely by the strong law of large numbers.
Glivenko and Cantelli strengthened this result by proving uniform convergence of Fn to F by the Glivenko–Cantelli theorem.
converges in distribution to a normal random variable N(0, P(A)(1 − P(A))) for fixed measurable set A.
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.
For real-valued iid random variables X1, X2, ..., Xn they are given by In this case, empirical processes are indexed by a class