Summary
Let μ n →μ be a weakly converging sequence of Borel probability measures on a topological space X. We prove the existence of an almost surely converging sequence of random variables ξ n →ξ which obey this laws, if a certain μ-dependent countability property of the topology holds. Especially this is the case if
-
(a)
X is second countable
-
(b)
X is first countable and μ has countable support
-
(c)
X is metrizable and μ is τ-smooth.
A final example disproves the existence of such random variables for (tight) measures on a Lusin space.
Article PDF
Similar content being viewed by others
References
Beran, R.J., Le Cam, L., Millar, P.W.: Convergence of stochastic empirical measures. J. Multivariate Anal. 23, 159–168 (1987)
Dudley, R.M.: Distances of probability measures and random variables. Ann. Math. Stat. 39, 1563–1572 (1968)
Dudley, R.M.: An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions. In: Lect. Notes Math. Vol. 1153, pp. 141–178. Berlin Heidelberg New York: Springer 1985
Fernandez, P.J.: Almost surely convergent versions of sequences which converge weakly. Bol. Soc. Bras. Mat. 5, 51–61 (1974)
Fernique, X.: Un modèle presque sûr pour la convergence en loi. C.R. Acad. Sci. Paris, t.306, Série I, 335–338 (1988)
Halmos, P.R.: Measure theory. New York: van Nostrand 1950
Hoffmann-Jørgensen, J.: The general marginal problem. In: Lect. Notes Math., Vol. 1242, pp. 77–367. Berlin Heidelberg New York: Springer 1987
Pyke, R.: Applications of almost surely convergent constructions of weakly convergent processes. In: Lect. Notes Math., Vol. 89, pp. 187–200. Berlin Heidelberg New York: Springer 1969
Schief, A.: On continuous image averaging of Borel measures. To appear in topology and its applications
Schief, A.: Topological properties of the addition map in spaces of Borel measures. Math. Ann. 282, 23–31 (1988)
Schwartz, L.: Radon measures. London: Oxford University Press 1973
Skorokhod, A.V.: Limit theorems for stochastic processes. Theory Probab. Appl. 1, 261–290 (English), 289–319 (Russian) (1956)
Steen, L., Seebach, J.: Counterexamples in topology, 2. ed. New York: Springer 1978
Topsøe F.: Topology and measure (Lect. Notes Math., Vol. 133). Berlin Heidelberg New York: Springer 1970
Wichura, M.: On the construction of almost uniformly convergent random variables with given weakly convergent image laws. Ann. Math. Stat. 41, 284–291 (1970)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Schief, A. Almost surely convergent random variables with given laws. Probab. Th. Rel. Fields 81, 559–567 (1989). https://doi.org/10.1007/BF00367303
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00367303