Empirical Process

An empirical process is a stochastic object in probability theory and statistics that characterizes the deviation of the empirical distribution function, the distribution built directly from a sample of data, from the true distribution it is estimating. For independent, identically distributed random variables, the empirical distribution function converges to the true distribution almost surely as the sample grows, a result strengthened by Glivenko and Cantelli, who proved that this convergence is uniform. Scaled versions of empirical processes behave approximately like normal distributions, extending the classical central limit theorem to a much broader setting, and Donsker's theorem shows that empirical processes indexed by suitable classes of functions converge weakly to Gaussian processes, giving rise to what are called Donsker classes. Empirical processes provide the theoretical foundation for non-parametric statistics, the branch of statistics that draws conclusions from data without assuming it follows a specific distribution.