The 1971 Wald Memorial Lectures

Distribution Function Inequalities for Martingales
Part of the Selected Works in Probability and Statistics book series (SWPS)


Let uf and Vf be nonnegative random variables associated with a martingale ƒ. In many interesting cases, the inequality
$$P\left( {Vf >\lambda } \right) \leqq cP\left( {Uf >\lambda } \right),$$
which usually does not hold for all λ > 0, does hold for enough λ so that
$$EVf \leqq cEUf$$
and more. The underlying theory, introduced in [6], has also proved fruitful in other probability applications; see [5] and [8]. For an entirely nonprobabilistic application to harmonic functions, see [7].


Maximal Function Sample Function Nonnegative Random Variable Nonnegative Measurable Function Walsh Series 


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  1. 1.
    Austin, D. G. (1966). A sample function property of martingales. Ann. Math. Statist. 371396-1397.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    BurkholderD. L. (1966). Martingale transforms. Ann. Math. Statist. 371494-1504.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    BurkholderD. L. (1968). Independent sequences with the Stein property. Ann. Math. Statist. 391282-1288.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    BurkholderD. L. (1971). Inequalities for operators on martingales. Proc. Internat. Congr. Math.(Nice, 1970). 2551-557. Gauthier-Villars, Paris.MathSciNetGoogle Scholar
  5. 5.
    BurkholderD. L., DavisB. J. and GundyR. F. (1972). Integral inequalities for convex functions of operators on martingales. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 2223-240.MathSciNetGoogle Scholar
  6. 6.
    BurkholderD. L. and GundyR. F. (1970). Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124249-304.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    BurkholderD. L. and GundyR. F. (1972). Distribution function inequalities for the area integral. Studia Math. 44117-134.MathSciNetGoogle Scholar
  8. 8.
    BurkholderD. L., GundyR. F. and SilversteinM. L. (1971). A maximal function characterization of the class Hp. Trans. Amer. Math. Soc. 157137-153.MATHMathSciNetGoogle Scholar
  9. 9.
    ChowY. S. (1967). On a strong law of large numbers for martingales. Ann. Math. Statist. 38610.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    ChowY. S. (1968). Convergence of sums of squares of martingale differences. Ann. Math. Statist. 39123-133.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    ChowY. S. (1971). On the Lp-convergence for n - 1 /S n, 0 < p <2. Ann. Math. Statist. 42393-394.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Davis, Burgess (1969). A comparison test for martingale inequalities. Ann. Math. Statist. 40505-508.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Davis, Burgess (1970). On the integrability of the martingale square function. Israel J. Math. 8187-190.CrossRefMathSciNetGoogle Scholar
  14. 14.
    Doleans, Catherine (1969). Variation quadratique des martingales continues à droite. Ann. Math. Statist. 40284-289.CrossRefMathSciNetGoogle Scholar
  15. 15.
    DoobJ. L. (1953). Stochastic Processes. Wiley, New York.MATHGoogle Scholar
  16. 16.
    DoobJ. L. (1954). Semimartingales and subharmonic functions. Trans. Amer. Math. Soc. 7786-121.MATHMathSciNetGoogle Scholar
  17. 17.
    FeffermanC. and SteinE. M. (1972). H pspaces of several variables. Acta Math. 129137-194.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Garsia, Adriano M. Recent progress in the theory of martingale inequalities. To appear.Google Scholar
  19. 19.
    Garsia, Adriano M. (1973). On a convex function inequality for martingales. Ann. Proba bility 1 .171-174.MATHCrossRefGoogle Scholar
  20. 20.
    GetoorR. K. and SharpeM. J. (1972). Conformai martingales. Invent. Math. 16271-308.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Gordon, Louis (1972). An equivalent to the martingale square function inequality. Ann. Math. Statist. 431927-1934.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Gundy, Richard F. (1968). A decomposition for L 1-bounded martingales. Ann. Math. Statist. 39134-138.CrossRefMathSciNetGoogle Scholar
  23. 23.
    Gundy, Richard F. (1969). On the class Llog L, martingales, and singular integrals. Studia Math. 33109-118.MathSciNetGoogle Scholar
  24. 24.
    Herz, Carl S. Bounded mean oscillation and regulated martingales. Canad. J. Math.To appear.Google Scholar
  25. 25.
    Hunt, Richard A. (1971). Almost everywhere convergence of Walsh-Fourier series of L 2functions. Proc. Internat. Congr. Math.(Nice, 1970). 2655-661. Gauthier-Villars, Paris.Google Scholar
  26. 26.
    JohnF. and NirenbergL. (1961). On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14415-426.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    MarcinkiewiczJ. and ZygmundA. (1938). Quelques théorèmes sur les fonctions in dépendantes. Studia Math. 7104-120.MATHGoogle Scholar
  28. 28.
    McKeanH. P., Jr. (1969). Stochastic Integrals. Academic Press, New York.MATHGoogle Scholar
  29. 29.
    MeyerP. A. (1969). Les inégalités de Burkholder en théorie des martingales. Séminaire de Probabilités III(Univ. Strasbourg, 1967/68) 163-174. Springer, Berlin.Google Scholar
  30. 30.
    MeyerP. A. (1972). Martingales and Stochastic IntegralsI. Springer, Berlin.MATHGoogle Scholar
  31. 31.
    MeyerP. A. (1973). Le dual de "H l "est "BMO" (cas continu). Séminaire de ProbabilitiesVII (Univ. Strasbourg, 1971/72). To appear. Springer, Berlin.Google Scholar
  32. 32.
    MillarP. Warwick (1968). Martingale integrals. Trans. Amer. Math. Soc.133 145-166.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Neveu, Jacques. Forthcoming book.Google Scholar
  34. 34.
    NovikovA. A. (1971). On moment inequalities for stochastic integrals. (Russian. English summary) Teor. Verojatnost. i Primenen. 16548-551.Google Scholar
  35. 35.
    PaleyR. E. A. C. (1932). A remarkable series of orthogonal functions I. Proc. London Math. Soc. 34241-264.MATHCrossRefGoogle Scholar
  36. 36.
    Rao, Murali (1972). Doob decomposition and Burkholder inequalities. Séminaire de Prob abilités VI(Univ. Strasbourg, 1970/71) 198-201. Springer, Berlin.Google Scholar
  37. 37.
    Rosenkrantz, Walter A. (1967). On rates of convergence for the invariance principle. Trans. Amer. Math. Soc. 129542-552.CrossRefMathSciNetGoogle Scholar
  38. 38.
    Rosenkrantz, Walter, and Sawyer, Stanley (1972). An elementary derivation of the moment inequalities for the Skorokhod stopping times. Preprint.Google Scholar
  39. 39.
    Rosenthal, Haskell P. (1970). On the subspaces of L p(p > 2) spanned by sequences of independent random variables. Israel J. Math. 8273-303.CrossRefMathSciNetGoogle Scholar
  40. 40.
    SawyerS. (1967). A uniform rate of convergence for the maximum absolute value of partial sums in probability. Comm. Pure Appl. Math. 20647-658.MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Sjölin, Per (1968). An inequality of Paley and convergence a.e. of Walsh-Fouries series. Ark. Mat. 7551-570.Google Scholar
  42. 42.
    Stein, Elias M. (1970). Topics in harmonic analysis related to the Littlewood-Paley theory. Ann. of Math. Studies 63.Google Scholar
  43. 43.
    Zygmund A. (1959). Trigonometric Series I.Cambridge Univ. Press.Google Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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