Mosco Convergence and Large Deviations

  • S. L. Zabell
Chapter
Part of the Progress in Probability book series (PRPR, volume 30)

Abstract

The techniques of convex analysis have come to play an increasingly important role in the theory of large deviations (see, e.g., Bahadur and Zabell, 1979; Ellis, 1985; de Acosta, 1988). The purpose of this brief note is to point out an interesting connection between a basic form of convergence commonly employed in convex analysis (“Mosco convergence”), and two theorems of fundamental importance in the theory of large deviations.

Keywords

Entropy 

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References

  1. [1]
    de Acosta, A. (1985). Upper bounds for large deviations of dependent random vectors. Z. Wahrscheinlichkeitstheorie verw. Gebiete. 69, 551–564.MATHCrossRefGoogle Scholar
  2. [2]
    de Acosta, A. (1988). Large deviations for vector-valued functionals of a Markov chain: lower bounds. Ann. Probab. 16, 925–960.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Bahadur, R. R., and Zabell, S. L. (1979). Large deviations of the sample mean in general vector spaces. Ann. Probab. 7, 587–621.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Baum, L.E, Katz, M., and Read, R. R. (1962). Exponential convergence rates for the law of large numbers. Trans. Amer. Math. Soc. 102, 187–199.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Dinwoodie, I. H. and Zabell, S. L. (1992). Large deviations for exchangeable sequences. Ann. Probab. 20. In press.Google Scholar
  6. [6]
    Dinwoodie, I. H. and Zabell, S. L. (1993). Large deviations for sequences of mixtures. Festschrift in honor of R. R. Bahadur. To appear.Google Scholar
  7. [7]
    Ellis, R. S. (1984). Large deviations for a general class of random vectors. Ann. Probab. 12, 1–16.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Springer-Verlag, New York.MATHCrossRefGoogle Scholar
  9. [9]
    Gärtner, J. (1977). On large deviations from the invariant measure. Theory Probab. Appl. 22, 24–39.MATHCrossRefGoogle Scholar
  10. [10]
    Lynch, J. (1978). A curious converse of Siever’s theorem. Ann. Probab. 6, 169–173.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Mosco, U. (1971). On the continuity of the Young-Fenchel transform. J. Math. Analysis Appl. 35, 518–535.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Plachky, D. (1971). On a theorem of G. L. Sievers. Ann. Math. Statist. 42, 1442–1443.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Plachky, D. and Steinebach, J. (1975). A theorem about probabilities of large deviations with applications to queuing theory. Period. Math. Hungar. 5, 343–345.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Sievers, G. L. (1969). On the probabilities of large deviations and exact slopes. Ann. Math. Statist. 40, 1908–1921.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Steinebach, J. (1978). Convergence rates of large deviation probabilities in the multidimensional case. Ann. Probab. 6, 751–759.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Zabell, S. L. (1992). Mosco convergence in locally convex spaces. J. Funct. Analysis. To appear.Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • S. L. Zabell
    • 1
  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA

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