Cramér’s Theorem in Banach Spaces Revisited

  • Pierre Petit
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)


The text summarizes the general results of large deviations for empirical means of independent and identically distributed variables in a separable Banach space, without the hypothesis of exponential tightness. The large deviation upper bound for convex sets is proved in a nonasymptotic form; as a result, the closure of the domain of the entropy coincides with the closed convex hull of the support of the common law of the variables. Also a short original proof of the convex duality between negentropy and pressure is provided: it simply relies on the subadditive lemma and Fatou’s lemma, and does not resort to the law of large numbers or any other limit theorem. Eventually a Varadhan-like version of the convex upper bound is established and embraces both results.


Cramér’s theory Large deviations Subadditivity Convexity Fenchel-Legendre transformation 

MSC 2010 Subject Classifications




I would like to thank Raphaël Cerf and Yann Fuchs for their careful reading, and the referee for his suggestions.


  1. 1.
    R. Azencott, Grandes déviations et applications, in École d’Été de Probabilités de Saint-Flour VIII-1978. Lecture Notes in Mathematics, vol. 774 (Springer, Berlin, 1980)Google Scholar
  2. 2.
    R.R. Bahadur, Some Limit Theorems in Statistics (SIAM, Philadelphia, 1971)Google Scholar
  3. 3.
    R.R. Bahadur, R. Ranga Rao, On deviations of the sample mean. Ann. Math. Stat. 31(4), 1015–1027 (1960)Google Scholar
  4. 4.
    R.R. Bahadur, S.L. Zabell, Large deviations of the sample mean in general vector spaces. Ann. Probab. 7(4), 587–621 (1979)Google Scholar
  5. 5.
    P. Bártfai, Large deviations of the sample mean in Euclidean spaces. Mimeograph Series No. 78-13, Statistics Department, Purdue University (1978)Google Scholar
  6. 6.
    P. Billingsley, Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics (Wiley, New York, 1999)Google Scholar
  7. 7.
    A.A. Borovkov, B.A. Rogozin, О щентральной прещельной теореме в многомерном случае. Teor. Verojatnost. i Primenen. 10(1), 61–69 (1965). English translation: On the multi-dimensional central limit theorem. Theory Probab. Appl. 10(1), 55–62 (1965)Google Scholar
  8. 8.
    R. Cerf, On Cramér’s theory in infinite dimensions. Panoramas et Synthèses 23. Société Mathématique de France, Paris (2007)Google Scholar
  9. 9.
    R. Cerf, P. Petit, A short proof of Cramér’s theorem in \(\mathbb {R}\). Am. Math. Mon. 118(10), 925–931 (2011)Google Scholar
  10. 10.
    H. Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23, 493–507 (1952)Google Scholar
  11. 11.
    H. Cramér, Sur un nouveau théorème-limite de la théorie des probabilités. Actual. Sci. Indust. 736, 5–23 (1938)Google Scholar
  12. 12.
    A. De Acosta, On large deviations of sums of independent random vectors, in Probability in Banach Spaces V. Lecture Notes in Mathematics, vol. 1153 (Springer, Berlin, 1985)Google Scholar
  13. 13.
    A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, 2nd edn. (Springer, New York, 1998). First edition by Jones and Bartlett in 1993Google Scholar
  14. 14.
    J.D. Deuschel, D.W. Stroock, Large Deviations (Academic, New York, 1989)Google Scholar
  15. 15.
    I.H. Dinwoodie, Identifying a large deviation rate function. Ann. Probab. 21(1), 216–231 (1993)Google Scholar
  16. 16.
    M.D. Donsker, S.R.S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time III. Commun. Pure Appl. Math. 29, 389–461 (1976)Google Scholar
  17. 17.
    N. Dunford, J.T. Schwartz, Linear Operators. Part I: General Theory (Wiley, New York, 1958)Google Scholar
  18. 18.
    M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit. ganzzahligen Koeffizienten. Math. Z. 17, 228–249 (1923)Google Scholar
  19. 19.
    J.M. Hammersley, Postulates for subadditive processes. Ann. Probab. 2(4), 652–680 (1974)Google Scholar
  20. 20.
    A.B. Hoadley, On the probability of large deviations of functions of several empirical cdf’s. Ann. Math. Stat. 38, 360–381 (1967)Google Scholar
  21. 21.
    W. Hoeffding, On probabilities of large deviations, in Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, 1965/1966. Vol. I: Statistics (University of California Press, Berkeley, 1967), pp. 203–219Google Scholar
  22. 22.
    J.F.C. Kingman, Subadditive Processes, in École d’Été de Probabilités de Saint-Flour V-1975. Lecture Notes in Mathematics, vol. 539 (Springer, Berlin, 1976)Google Scholar
  23. 23.
    O.E. Lanford, Entropy and equilibrium states in classical statistical mechanics, in Statistical Mechanics and Mathematical Problems. Lecture Notes in Physics, vol. 20 (Springer, Berlin, 1973)Google Scholar
  24. 24.
    J.T Lewis, C.E. Pfister, W.G. Sullivan, Entropy, concentration of probability and conditional limit theorems. Markov Processes Relat. Fields 1(3), 319–386 (1995)Google Scholar
  25. 25.
    J.J. Moreau, Fonctionnelles convexes. Séminaire sur les Équations aux Dérivées Partielles, Collège de France (1966–1967)Google Scholar
  26. 26.
    K.R. Parthasarathy, Probability Measures on Metric Spaces (AMS Chelsea Publishing, Providence, 2005). Reprint of the 1967 originalGoogle Scholar
  27. 27.
    V.V. Petrov, О вероятностях больших уклонений сумм независимюх случайнюх величин. Teor. Verojatnost. i Primenen 10(2), 310–322 (1965). English translation: On the probabilities of large deviations for sums of independent random variables. Theory Probab. Appl. 10(2), 287–298 (1965)Google Scholar
  28. 28.
    W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973)zbMATHGoogle Scholar
  29. 29.
    D. Ruelle, Correlation functionals. J. Math. Phys. 6(2), 201–220 (1965)MathSciNetCrossRefGoogle Scholar
  30. 30.
    I.N. Sanov, О вероятности больших отклонений случайнюх величин. Mat. Sb. 42(1), 11–44 (1957). English translation: On the probability of large deviations of random variables. Sel. Transl. Math. Statist. Probab. I, 213–244 (1961)Google Scholar
  31. 31.
    J. Sethuraman, On the probability of large deviations of families of sample means. Ann. Math. Stat. 35, 1304–1316 (1964)MathSciNetCrossRefGoogle Scholar
  32. 32.
    J. Sethuraman, On the probability of large deviations of the mean for random variables in D[0, 1]. Ann. Math. Stat. 36, 280–285 (1965)MathSciNetCrossRefGoogle Scholar
  33. 33.
    G.L. Sievers, Multivariate probabilities of large deviations. Ann. Stat. 3(4), 897–905 (1975)MathSciNetCrossRefGoogle Scholar
  34. 34.
    S.R.S. Varadhan, Asymptotic probabilities and differential equations. Commun. Pure Appl. Math. 19, 261–286 (1966)MathSciNetCrossRefGoogle Scholar
  35. 35.
    C. Zălinescu, Convex Analysis in General Vector Spaces (World Scientific, River Edge, 2002)CrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseUPS IMTToulouse CedexFrance

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