Compactness criteria for the stable topology

• Autorzy: Raynaud de Fitte P.
• Języki publikacji: EN

Abstrakty

EN

Adapting a general result of Topsoe [32], we prove a compactness criterion for the stable topology on the set of measures on the product of a measurable space and a Suslin (nonnecessarily regular) topological space. We also extend a compactness criterion of Jacod and Memin [17] and we apply these results to the case of Young measures.

Słowa kluczowe

EN

stable convergence; Young measures; narrow topology; semicontinuity theorem

PL

zbieżność stabilna; miara Younga; twierdzenie półciągłości

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2003
• Tom: Vol. 51, no 4
• Strony: 343--363
• Opis fizyczny: Bibliogr. 40 poz.

Twórcy

autor

Raynaud de Fitte P.

• Laboratoire de Mathématiques Raphaël Salem, UMR CNRS 6085, Site Colbert, Faculté des Sciences, Université de Rouen, 76 821 Mont-Saint-Aignan Cedex, France

Bibliografia

• [1] W. Adamski, P. Gänssler, S. Kaiser, On compactness and convergence in spaces of measures, Math. Ann., 220 (3) (1976) 193-210.
• [2] E. Balder, On Prohorov’s theorem for transition probabilities, Sém. Anal. Convexe, 19 (1989) 9.1-9.11.
• [3] E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., 22 (1984) 570-598.
• [4] E. J. Balder, Lectures on Young measure theory and its applications in economics, Rend. Istit. Mat. Univ. Trieste, 31 (2000) suppl.: 1-69. Workshop di Teoria della Misura et Analisi Reale Grado, 1997 (Italia).
• [5] E. J. Balder, On ws-converqence of product measures, Math. Oper. Res., 26 (3) (2001) 494-518.
• [6] V. I. Bogachev, Measures on topological spaces, J. Math. Sci., 91 (4) (1998) 3033-3156.
• [7] N. Bourbaki, Espaces vectoriels topologiques, second edition, Masson, Paris 1981.
• [8] G. Buttazzo, Semicontinuity, relaxation and integral representation in the calculus of variations, Pitman Res. Notes Math. Ser., no. 207 (1989).
• [9] C. Castaing, M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Math., no. 580, Springer Verlag, Berlin 1977.
• [10] C. Dellacherie, P. A. Meyer, Probabilités et potentiel. Chapitres I à IV, Hermann, Paris 1975.
• [11] C. Dellacherie, P. A. Meyer, Probabilités et potentiel. Chapitres IX à XI, Théorie discrète du potentiel, Hermann, Paris 1983.
• [12] R. M. Dudley, Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Cole, Pacific Grove, CA 1989.
• [13] R. Engelking, General topology, Heldermann Verlag, Berlin 1989.
• [14] F. Galdéano, Convergence étroite de mesures définies sur un espace produit, Ph.D. Thesis, Université de Perpignan, 1997.
• [15] P. Gänssler, Compactness and sequential compactness in spaces of measures, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 17 (1971) 124-146.
• [16] J. Jacod, On continuous conditional Gaussian martingales and stable convergence in law, in: Séminaire de Probababilités XXXI, Lecture Notes in Math., no. 1655, Springer Verlag, Berlin (1997) 232-246.
• [17] J. Jacod, J. Mémin, Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité, in: Sémin. de Probababilités XV, Univ. Strasbourg 1979/80, Lecture Notes in Math., no. 850, Springer Verlag, Berlin (1981) 529-546.
• [18] J. Jacod, J. Mémin, Rectification à “Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité”, in: Sémin. de Probababilités XVII, proc. 1981/82, Lecture Notes in Math., no. 986, Springer Verlag, Berlin (1983) 509-511.
• [19] A. Jawhar, Mesures de transition et applications, Sém. Anal. Convexe, 14 (1984) 13.1-13.62.
• [20] J. L. Kelley, General topology. Springer Verlag, Berlin 1955.
• [21] G. Letta, Convergence stable et applications, Atti Sem. Mat Fis. Modena (supplemento), 46 (1998) 191-211.
• [22] G. L. O’Brien, S. Watson, Relative compactness for capacities, measures, upper semicontinuous functions and closed sets, J. Theoret. Probab., 11 (1998) 577-588.
• [23] K. R. Parthasarathy, Probability measures on metric spaces, Acad. Press, New York, London 1967.
• [24] J. Pellaumail, Solutions faibles et semi-martingales, in: Séminaire de Probababilités XV, 1979/80. Université de Strasbourg, Lecture Notes in Math., Springer Verlag, Berlin (1981) 561-586.
• [25] L. Piccinini, M. Valadier, Uniform integrability and Young measures, J. Math. Anal. Appl., 195 (2) (1995) 428-439.
• [26] A. Rényi, P. Revész, On mixing sequences of random variables, Acta Math. Acad. Sci. Hungar., 9 (1958) 389-393.
• [27] A. Rényi, On stable sequences of events, Sankhyā Ser. A, 25 (1963) 293-302.
• [28] T. Roubíček, Relaxation in optimization theory and variational calculus, de Gruyter Ser. Nonlinear Anal. Appl., (4) (1997).
• [29] M. Schäl, On dynamic programming: compactness of the space of policies, Stochastic Process. Appl., 3 (4) (1975) 345-364.
• [30] L. Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Tata Institute of Fundamental Research Studies in Mathematics, Oxford University Press, London 1973.
• [31] L. A. Steen, J. A. Seebach, Jr, Counterexamples in topology, Holt, Rinehart and Winston, Inc., New York 1978. Second edition. Dover Publications paperback 1996.
• [32] F. Topsøe, Compactness in spaces of measures, Studia Math., 36 (1970) 195-212.
• [33] F. Topsøe, Topology and measure, Lecture Notes in Math., no. 133, Springer Verlag, Berlin 1970.
• [34] F. Topsøe, Compactness and tightness in a space of measures with the topology of weak convergence, Math. Scand., 34 (1974) 187-210.
• [35] M. Valadier, Contribution à l’analyse convexe, Université de Montpellier, Montpellier 1970. (Thèse de Doctorat ès-Sciences - Mathématiques présentée à la Faculté des Sciences de Paris pour obtenir le grade de Docteur ès-Sciences. Secrétariat de Mathématiques de la Faculté de Montpellier, 1970, Publication no 92).
• [36] M. Valadier, Multiapplications mesurables à valeurs convexes compactes, J. Math. Pures Appl. (9), 50 (1971) 265-297.
• [37] M. Valadier, Désintégration d’une mesure sur un produit, C. R. Acad. Sci. Paris Sér. I, 276 (1973) A33-A35.
• [38] M. Valadier, A course on Young measures, Rend. Istit. Mat. Univ. Trieste, 26 (1994) suppl.: 349-394. Workshop di Teoria della Misura et Analisi Reale Grado, 1993 (Italia).
• [39] M. Valadier, Analysis of the asymptotic distance between oscillating functions and their weak limit in L2, Adv. Math. Econ., 1 (1999) 99-113.
• [40] L. C. Young, Generalized curves and the existence of an attained absolute minimum in the Calculus of Variations, C. R. Soc. Sc. Varsovie, 30 (1937) 212-234.