Abstract
Assume that a net (σα) of measures converges in some sense to a measureΜ. Then we investigate whether for a given class ℰ of functions, we can conclude that
.
The present paper offers an axiomatic treatment which allows us to extend the “ζ-criterion”, until now only available in the weak convergence case, to the case of setwise convergence as well. Actually this was our principal motivation.
The paper is selfcontained and yet due to the development of new techniques there is only little overlap with previous research.
Some concrete results — mainly refinements of known ones — are derived with the present theory at hand. In particular, we characterize those probability distributions onR n for which the empirical distributions almost surely converge to the theoretical one, uniformly over the class of convex sets.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahmad, S.: Sur le Théorème de Glivenko-Cantelli. C.R. Acad. Sci. Paris Sér. A–B252, 1413–1417 (1961)
Billingsley, P., TopsØe, F.: Uniformity in weak convergence. Z. Wahrscheinlichkeitstheorie verw. Gebiete7, 1–16 (1967)
Dunford, N., Schwartz, J.T.: Linear operators: Part I, general theory. New York: Interscience publishers 1963
Elker, J.: Unpublished “Diplomarbeit” from the Ruhr university, Bochum 1975
Engelking, R.: Outline of general topology. Amsterdam: North-Holland 1968
Fabian, V.: On uniform convergence of measures. Z. Wahrscheinlichkeitstheorie verw. Gebiete15, 139–143 (1970)
Gaenssler, P.: On convergence of sample distributions. Bull. Inst. Internat. Statist., Proc. 39th Session45, 427–432 (1973)
Gaenssler, P., Stute, W.: On uniform convergence of measures with applications to uniform convergence of empirical distributions. In Empirical Distributions and Processes. Lecture notes in mathematics, 566. Berlin-Heidelberg-New York: Springer 1976
Halmos, P.R.: Mesure theory. Princeton: D. Van Nostrand 1950
Bhaskara Rao, M., Bhaskara Rao, K.P.S.: Borelσ-algebra on [0,Ω]. Manuscripta math.5, 195–198 (1971)
Ranga Rao, R.: Relations between weak and uniform convergence of measures with applications. Ann. math. Statistics33, 659–680 (1962)
Ressel, P.: Some continuity and measurability results on spaces of measures. Copenhagen University Preprint Series No 12, 1976
Rockafellar, R.T.: Convex analysis. Princeton: Princeton University Press 1970
Stute, W.: On a generalization of the Glivenko-Cantelli Theorem. Z. Wahrscheinlichkeitstheorie verw. Gebiete35, 167–175 (1976)
Stute, W.: A necessary condition for the convergence of the isotrope discrepancy. In Empirical Distributions and Processes. Lecture notes in mathematics, 566. Berlin-Heidelberg-New York: Springer 1976
Stute, W.: On uniformity classes of functions with an application to the speed of Mean Glivenko-Cantelli convergence. Ruhr-University Bochum, Preprint Series No. 5, 1974
TopsØe, F.: On the connection betweenP-continuity andP-uniformity in weak convergence. Theor. Probability Appl.12, 281–290 (1967)
TopsØe, F.: On the Glivenko-Cantelli theorem. Z. Wahrscheinlichkeitstheorie verw. Gebiete14, 239–250 (1970)
TopsØe, F.: Topology and measure. Lecture Notes in Math. 133. Berlin-Heidelberg-New York: Springer 1970
Added Reference
Eddy, W.F., Hartigan, J.A.: Uniform convergence of the empirical distribution function over convex sets, to appear in the Annals of Statistics
Author information
Authors and Affiliations
Additional information
Research supported by the Danish Natural Science Research Council
Rights and permissions
About this article
Cite this article
Topsøe, F. Uniformity in convergence of measures. Z. Wahrscheinlichkeitstheorie verw Gebiete 39, 1–30 (1977). https://doi.org/10.1007/BF01844870
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01844870