This article is cited in 6 papers

Short Communications

On the Glivenko–Cantelli Theorem

V. V. Sazonov

Moscow

Abstract: It is found that the Glivenko–Cantelli theorem does not admit a generalisation to the infinite dimensional case. In fact under general conditions, with probability one the empirical distributions do not converge to the theoretical distribution uniformly on all half-spaces. Sufficient conditions for the validity of uniform convergence over a family of half-spaces with probability one are given.

Received: 24.02.1962

 English version: Theory of Probability and its Applications, 1963, 8:3, 282–286