Summary
In a separable metric space, if two Borel probability measures (laws) are nearby in a suitable metric, then there exist random variables with those laws which are nearby in probability. Specifically, by a well-known theorem of Strassen, the Prohorov distance between two laws is the infimum of Ky Fan distances of random variables with those laws. The present paper considers possible extensions of Strassen's theorem to two random elements one of which may be (compact) set-valued and/or non-measurable. There are positive results in finite-dimensional spaces, but with factors depending on the dimension. Examples show that such factors cannot entirely be avoided, so that the extension of Strassen's theorem to the present situation fails in infinite dimensions.
Article PDF
Similar content being viewed by others
References
[A] Andersen, N.T.: The central limit theorem for non-separable valued functions. Z. Wahrscheinlichkeitstheor. Verw. Geb.70, 445–455 (1985)
[B] Banach, S.: Théorie des opérations linéaires. Warsaw, 1932; Repr. Chelsea, New York (1955)
[D] Dudley, R.M.: Real analysis and probability. 2d printing, corrected. New York London. Chapman and Hall (1993)
[D-P] Dudley, R.M., Philipp, W.: Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.62, 509–552 (1983)
[F-L-M] Figiel, T., Lindenstrauss, J., Milman, V.D.: The dimension of almost spherical sections of convex bodies. Acta Math. (Sweden)139 53–94 (1977)
[G-Z] Giné, E., Zinn, J.: Gaussian characterization of uniform Donsker classes of functions. Ann. Probab.19, 758–782 (1991)
[K-R] Kantorovich, L.V., Rubinštein, G.Š.: On a space of completely additive functions. Vestnik Leningrad University1958 no. 7 (Ser. Mat. Mekh. Astron. vyp. 2, 52–59) (in Russian)
[L1] Lévy, Paul: Lecons d'analyse fonctionnelle. Paris: Gauthier-Villars 1922
[L2] L'evy, Paul: Problèmes concrets d'analyse fonctionelle (2d ed. of Lévy, 1922) Paris: Gauthier-Villars 1951
[M] Milman, V.D.: The heritage of P. Lévy in geometrical functional analysis. Astérisque157–158, 273–301 (1988)
[M-S] Milman, V.D., Schechtman, G.: Asymptotic theory of finite dimensional normed spaces, with an Appendix by M. Gromov (Lect. Notes Math., vol. 1200) Berlin Heidelberg New York: Springer 1986
[N] Nachbin, L.: The Haar integral. New York: Van Nostrand 1965
[O] Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc.84, 1182–1238 (1978)
[R] Rachev, S.T.: The Monge-Kantorovich mass transference problem and its stochastic applications. Theory Probab. Appl.29, 647–676 (1985)
[R-R-S] Rachev, S.T., Rüschendorf, L., Schief, A.: Uniformities for the convergence in law and in probability. J. Theor. Probab.5, 33–44 (1992)
[S1] Schmidt, E.: Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sphärischen Raum jeder Dimensionenzahl. Math. Z.49, 1–109 (1943)
[S2] Schmidt, E.: Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie I, II. Math. Nachr.1, 81–157;2, 171–244 (1948, 1949)
[St] Strassen, Volker: The existence of probability measures with given marginals. Ann. Math. Stat.36, 423–439 (1965)
[T] Taylor, A.E.: A geometric theorem and its application to biorthogonal systems. Bull. Am. Math. Soc.53, 614–616 (1947)
Author information
Authors and Affiliations
Additional information
This research was partially supported by a Guggenheim Fellowship, by National Science Foundation grant DMS 8505550 at MSRI-Berkeley, and other NSF grants