Abstract
We give a survey of recent results that generalize and develop a classical theorem of Skorokhod on representation of weakly convergent sequences of probability measures by almost everywhere convergent sequences of mappings.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
A. V. Skorokhod, “Limit theorems for stochastic processes,” Theory Probab. Appl., 1, No.3, 261–290 (1956).
A. V. Skorokhod, Studies in the Theory of Random Processes, Addison-Wesley, Reading (1965).
D. Blackwell and L. E. Dubins, “An extension of Skorokhod's almost sure representation theorem,” Proc. Amer. Math. Soc., 89, No.4, 691–692 (1983).
R. M. Dudley, “Distances of probability measures and random variables,” Ann. Math. Statist., 39, 1563–1572 (1967).
X. Fernique, “Un modele presque sur pour la convergence en loi,” C. R. Acad. Sci. Paris. Ser. 1, 306, 335–338 (1988).
A. Jakubowski, “The almost sure Skorokhod representation for subsequences in nonmetric spaces,” Theory Probab. Appl., 42, No.1, 167–174 (1997).
G. Letta and L. Pratelli, “Le theoreme de Skorokhod pour des lois de Radon sur un espace metrisable, ” Rend. Accad. Naz. XL Mem. Mat. Appl., 21, No.5, 157–162 (1997).
A. Schief, “Almost surely convergent random variables with given laws,” Probab. Theory Relat. Fields., 81, 559–567 (1989).
W. Szczotka, “A note on Skorokhod representation,” Bull. Pol. Acad. Sci. Math., 38, 35–39 (1990).
M. J. Wichura, “On the construction of almost uniformly convergent random variables with given weakly convergent image laws,” Ann. Math. Statist., 41, No.1, 284–291 (1970).
V. I. Bogachev and A. V. Kolesnikov, “Open mappings of probability measures and the Skorokhod representation theorem,” Theory Probab. Appl., 46, No.1, 1–21 (2001).
T. O. Banakh, V. I. Bogachev, and A. V. Kolesnikov, “Topological spaces with the strong Skorokhod property,” Georg. Math. J., 8, No.2, 201–220 (2001).
T. O. Banakh, V. I. Bogachev, and A. V. Kolesnikov, “On topological spaces with the Prokhorov and Skorokhod properties,” Dokl. Math., 64, No.2, 244–247 (2001).
T. O. Banakh, V. I. Bogachev, and A. V. Kolesnikov, “Topological spaces with the strong Skorokhod property, II,” in: Functional Analysis and Applications (Proceedings of the International Conference on Functional Analysis and Its Applications Dedicated to the 110th Anniversary of Stefan Banach, May 28–31, 2002, Lviv), Elsevier, Amsterdam (2004), pp. 23–47.
T. O. Banakh, V. I. Bogachev, and A. V. Kolesnikov, Monotone Compact-Covering Maps, Monotone ℵ-Spaces, and Their Applications in Probability Theory, Preprint, Bielefeld University, Bielefeld (2005).
V. I. Bogachev, Foundations of Measure Theory [in Russian], Vols. 1, 2, Moscow (2003.).
M. M. Choban, “Spaces, mappings, and compact subsets,” Bull. Acad. Sci. Rep. Moldova, No. 2(36), 3–52 (2001).
J. A. Cuesta-Albertos and C. Matran-Bea, “Stochastic convergence through Skorokhod representation theorems and Wasserstein distances,” First Int. Conf. Stochast. Geometry, Convex Bodies and Empirical Measures (Palermo, 1993), Rend. Circ. Mat. Palermo (2), Suppl., No. 35, 89–113 (1994).
A. Tuero, “On the stochastic convergence of representations based on Wasserstein metrics,” Ann. Probab., 21, No.1, 72–85 (1993).
A. N. Shiryaev, Probability, Springer (1990).
A. Pelczynski, Linear Extensions, Linear Averagings, and Their Applications to Linear Topological Classification of Spaces of Continuous Functions, Warszawa (1968).
A. Schief, “An open mapping theorem for measures,” Monatsh. Math., 108, No.1, 59–70 (1989).
P. Raynaud de Fitte, “Theoreme ergodique ponctuel et lois fortes des grands nombres pour des points aleatoires d'un espace metrique a courbure negative,” Ann. Probab., 25, No.2, 738–766 (1997).
V. A. Lebedev, Martingales, Convergence of Probability Measures, and Stochastic Equations [in Russian], MAI, Moscow (1996).
N. V. Krylov, “On SPDEs and superdiffusions,” Ann. Probab., 25, 1789–1809 (1997).
V. I. Bogachev, A. V. Kolesnikov, and K. V. Medvedev, “Triangular transformations of measures,” Sb. Math., 196, No.3 (2005).
R. D. Anderson, “Monotone interior dimension-raising mappings,” Duke Math. J., 19, 359–366 (1952).
P. S. Aleksandrov and B. A. Pasynkov, Introduction to Dimension Theory [in Russian], Nauka, Moscow (1973).
E. Michael, “A Selection Theorem,” Proc. Amer. Math. Soc., 17, 1404–1406 (1966).
D. Repovs and P. V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer, Dordrecht (1998).
V. V. Filippov, “On a question of E. A. Michael,” Comment. Math. Univ. Carol., 45, No.4, 735–738 (2004).
R. Engelking, General Topology, PWN, Warsaw (1977).
V. I. Bogachev, “Measures on topological spaces,” J. Math. Sci. (New York), 91, No.4, 3033–3156 (1998).
D. Fremlin, D. Garling, and R. Haydon, “Bounded measures on topological spaces,” Proc. London Math. Soc., 25, 115–136 (1972).
J. Diestel and J. J. Uhl, Vector Measures, American Mathematical Society, Providence, RI (1977).
I. V. Protasov, “Maximal topologies on groups,” Sib. Mat. Zh., 39, No.6, 1184–1194 (1998).
I. V. Protasov, “Extremal topologies on groups,” Mat. Stud., 15, No.1, 9–22 (2001).
P. Habala, P. Hajek, and V. Zizler, Introduction to Banach Spaces, Matfyzpress, Praha (1996).
A. V. Arkhangel'skii, Topological Function Spaces, Kluwer, Dordrecht (1992).
M. Fabian, Gateaux Differentiability of Convex Functions and Topology, Wiley-Interscience (1997).
R. Deville, G. Godefroy, and V. Zizler, Smoothness and Renorming in Banach Spaces, Longman, Essex (1993).
Author information
Authors and Affiliations
Additional information
__________
Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 9, pp. 1171–1186, September, 2005.
Rights and permissions
About this article
Cite this article
Banakh, T.O., Bogachev, V.I. & Kolesnikov, A.V. Topological Spaces with Skorokhod Representation Property. Ukr Math J 57, 1371–1386 (2005). https://doi.org/10.1007/s11253-006-0002-z
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11253-006-0002-z