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.
^Mojirsheibani, M. (2007). "Nonparametric curve estimation with missing data: A general empirical process approach". Journal of Statistical Planning and Inference. 137 (9): 2733–2758. doi:10.1016/j.jspi.2006.02.016.
Dzhaparidze, K. O.; Nikulin, M. S. (1982). "Probability distributions of the Kolmogorov and omega-square statistics for continuous distributions with shift and scale parameters". Journal of Soviet Mathematics. 20 (3): 2147. doi:10.1007/BF01239992. S2CID123206522.