Empirical processes

In probability theory, an empirical process is a 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 (without rescaling). 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.

For X₁, X₂, ... X_nindependent and identically-distributed random variables in R with common cumulative distribution function F(x), the empirical distribution function is defined by

where I_C is the indicator function of the set C.

For every (fixed) x, F_n(x) is a sequence of random variables which converge to F(x) almost surely by the strong law of large numbers. That is, F_n converges to F pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergence of F_n 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, ${\displaystyle G_{n}(A)}$ 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, ${\displaystyle G_{n}f}$ converges in distribution to a normal random variable ${\displaystyle N(0,\mathbb {E} (f-\mathbb {E} f)^{2})}$, provided that ${\displaystyle \mathbb {E} f}$ and ${\displaystyle \mathbb {E} f^{2}}$ exist.

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