Summary
LetI 2 be the unit cube of\(\mathbb{R}^2 \) andX i be independentI 2-valued random variables that are distributed according to Lebesgue-measure. IfS is the set of closed convex subsets ofI 2 we consider the processμ n (A) A∈S,
where\(\mu _n (A) = (1/n)\mathop \sum \limits^n 1{}_A(X_i ).\).It is proved that this process suitably normalized converges in a suitable weak sense to a Gaussian process.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bennet, G.: Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc,57, 33–45 (1962)
Billingsley, P.: Convergence of probability measures. New York: Wiley 1968
Dudley, R.M.: Weak convergence of probability measures on nonseparabel metric spaces and empirical measures on Euclidean spaces. Illinois J. Math.10, 109–126 (1968)
Dudley, R.M.: Sample functions of Gaussian processes. Ann. Probability1, 66–103 (1973)
Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundary. J. Approx. Th.10,227–236
deHoyos, A.: Continuity and convergence of some process parametrized by the compact sets inR s; Z.Wahrscheinlichkeitstheorie und verw. Gebiete23, 153–162 (1972)
Parthasaraty, K.R.: Probability measures on metric spaces. New York: Academic Press 1967
Strassen, V., Dudley, R.M.: The central limit theorem andɛ-entropy; Lecture notes in Mathematics89. Berlin-Heidelberg-New York: Springer 1969
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bolthausen, E. Weak convergence of an empirical process indexed by the closed convex subsets ofI 2 . Z. Wahrscheinlichkeitstheorie verw Gebiete 43, 173–181 (1978). https://doi.org/10.1007/BF00668458
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00668458