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 Electronic Communications in Probability > Vol. 13 (2008) > Paper 22 open journal systems 


Stone-Weierstrass type theorems for large deviations

Henri Comman, University of Santiago de Chile


Abstract
We give a general version of Bryc's theorem valid on any topological space and with any algebra $mathcal{A}$ of real-valued continuous functions separating the points, or any well-separating class. In absence of exponential tightness, and when the underlying space is locally compact regular and $mathcal{A}$ constituted by functions vanishing at infinity, we give a sufficient condition on the functional $Lambda(cdot)_{mid mathcal{A}}$ to get large deviations with not necessarily tight rate function. We obtain the general variational form of any rate function on a completely regular space; when either exponential tightness holds or the space is locally compact Hausdorff, we get it in terms of any algebra as above. Prohorov-type theorems are generalized to any space, and when it is locally compact regular the exponential tightness can be replaced by a (strictly weaker) condition on $Lambda(cdot)_{mid mathcal{A}}$.


Full text: PDF

Pages: 225-240

Published on: April 28, 2008
Bibliography
  1. Bentley, H. L.; Lowen-Colebunders, E. Completely regular spaces. Comment. Math. Univ. Carolin. 32 (1991), no. 1, 129--153. MR1118297 (93d:54028)
  2. Bryc, Wlodzimierz. On the large deviation principle for stationary weakly dependent random fields. Ann. Probab. 20 (1992), no. 2, 1004--1030. MR1159583 (93c:60031)
  3. Bryc, Wlodzimierz. Large deviations by the asymptotic value method. Diffusion processes and related problems in analysis, Vol. I (Evanston, IL, 1989), 447--472, Progr. Probab., 22, Birkhäuser Boston, Boston, MA, 1990. MR1110177 (93b:60055)
  4. Comman, Henri. Functional approach of large deviations in general spaces. J. Theoret. Probab. 18 (2005), no. 1, 187--207. MR2132276 (2006b:60041)
  5. Comman, Henri. Criteria for large deviations. Trans. Amer. Math. Soc. 355 (2003), no. 7, 2905--2923 (electronic). MR1975405 (2004d:60068)
  6. Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications.Second edition.Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2 MR1619036 (99d:60030)
  7. van Douwen, Eric K. A regular space on which every continuous real-valued function is constant. Nieuw Arch. Wisk. (3) 20 (1972), 143--145. MR0312462 (47 #1019)
  8. Dupuis, Paul; Ellis, Richard S. A weak convergence approach to the theory of large deviations.Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication.John Wiley & Sons, Inc., New York, 1997. xviii+479 pp. ISBN: 0-471-07672-4 MR1431744 (99f:60057)
  9. Eichelsbacher, Peter. Rate functions in the theory of large deviations. Statist. Probab. Lett. 26 (1996), no. 3, 279--283. MR1394904 (97f:60061)
  10. Engelking, R. Outline of general topology.Translated from the Polish by K. Sieklucki North-Holland Publishing Co., Amsterdam; PWN-Polish Scientific Publishers, Warsaw; Interscience Publishers Division John Wiley & Sons, Inc., New York 1968 388 pp. MR0230273 (37 #5836)
  11. Kelley, John L. General topology.Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.].Graduate Texts in Mathematics, No. 27.Springer-Verlag, New York-Berlin, 1975. xiv+298 pp. MR0370454 (51 #6681)
  12. Pedersen, Gert K. Analysis now.Graduate Texts in Mathematics, 118. Springer-Verlag, New York, 1989. xiv+277 pp. ISBN: 0-387-96788-5 MR0971256 (90f:46001)
  13. Steen, Lynn A.; Seebach, J. Arthur, Jr. Counterexamples in topology.Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London 1970 xiii+210 pp. MR0266131 (42 #1040)













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Electronic Communications in Probability. ISSN: 1083-589X