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Martingales

  • Yuan Shih Chow
  • Henry Teicher
Chapter
Part of the Springer Texts in Statistics book series (STS)

Abstract

An introduction to martingales appeared in Section 7.4, where convergence theorems for submartingales {Sn Sn, n ≥ 1} (relating to differentiation theory) were discussed. Here, emphasis will fall upon convergence theorems for martingales {S-n-nn ≤ -1} (relating to ergodic theorems). In demarcating the two cases, it is natural to refer to a martingale {Sn S nn ≥ 1} as an upward martingale and to allude to a martingale {S-n S -nn ≤ - 1} when written {Sn S n,n ≥ 1}as a downward or reverse martingale. Martingale and stochastic inequalities will also be dealt with.

Keywords

Nondecreasing Function Martingale Difference Uniform Integrability Martingale Difference Sequence Stochastic Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. D. G. Austin, “A sample property of martingales,”Ann. Math. Stat.37 (1966), 1396–1397.MathSciNetzbMATHCrossRefGoogle Scholar
  2. D. Blackwell, “On optimal systems,”Ann. Math. Stat.25 (1954), 394–397.MathSciNetzbMATHCrossRefGoogle Scholar
  3. B. M. Brown, “A note on convergence of moments,”Ann. Math. Stat.42 (1971), 777–779.zbMATHCrossRefGoogle Scholar
  4. D. L. Burkholder, “Martingale transforms,”Ann. Math. Stat.37 (1966), 1494–1504.MathSciNetzbMATHCrossRefGoogle Scholar
  5. D. L. Burkholder, “Distribution function inequalities for martingales,”Ann. Probability1 (1973), 19–42.MathSciNetzbMATHCrossRefGoogle Scholar
  6. D. L. Burkholder and R. F. Gundy, “Extrapolation and interpolation of quasi-linear operators on martingales,”Acta Math.124 (1970), 249–304.MathSciNetzbMATHCrossRefGoogle Scholar
  7. D. L. Burkholder, B. J. Davis, and R. F. Gundy, “ Inequalities for convex functions of operators on martingales,”Proc. Sixth Berkeley Symp. Math. Stat. Prob.2 (1972), 223–240.MathSciNetGoogle Scholar
  8. Y.S. Chow, “On a strong law of large numbers for martingales,”Ann. Math. Stat.38 (1967), 610.zbMATHCrossRefGoogle Scholar
  9. Y. S. Chow, “Convergence of sums of squares of martingale differences,”Ann. Math. Stat.39 (1968), 123–133.zbMATHCrossRefGoogle Scholar
  10. Y. S. Chow, “On the Le-convergence forn - 1 /PS„0<p <2,“Ann. Math. Stat.42 (1971), 393–394.zbMATHCrossRefGoogle Scholar
  11. K. L. ChungA Course in Probability TheoryHarcourt Brace, New York, 1968; 2nd ed., Academic Press, New York, 1974.Google Scholar
  12. B. Davis, “A comparison test for martingale inequalities,”Ann. Math. Stat.40 (1969), 505–508.zbMATHCrossRefGoogle Scholar
  13. J. L. DoobStochastic ProcessesWiley, New York, 1953.zbMATHGoogle Scholar
  14. L. E. Dubins and D. A. Freedman, “A sharper form of the Borel-Cantelli lemma and the strong law,”Ann. Math. Stat.36 (1965), 800–807.MathSciNetzbMATHCrossRefGoogle Scholar
  15. L. E. Dubins and L. J. SavageHow to Gamble If You MustMcGraw-Hill, New York, 1965.zbMATHGoogle Scholar
  16. A. M. Garsia, “On a convex function inequality for martingales,”Ann. Probability1 (1973), 171–174.MathSciNetzbMATHCrossRefGoogle Scholar
  17. R. F. Gundy, “A decomposition for 2’I-bounded martingales,”Ann. Math. Stat.39 (1968), 134–138.MathSciNetzbMATHCrossRefGoogle Scholar
  18. J. NeveuMartingales à temps discretsMasson, Paris, 1972.Google Scholar
  19. R. Panzone, “Alternative proofs for certain uperossing inequalities,”Ann. Math. Stat.38 (1967), 735–741.MathSciNetzbMATHCrossRefGoogle Scholar
  20. E. M. SteinTopics in Harmonic Analysis Related to the Littlewood—Paley TheoryPrinceton Univ. Press, Princeton, 1970.zbMATHGoogle Scholar
  21. H. Teicher, “Moments of randomly stopped sums-revisited,”Jour. Theor. Prob.9 (1995), 779–793.MathSciNetCrossRefGoogle Scholar
  22. A. ZygmundTrigonometric SeriesVol. I, Cambridge, 1959.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Yuan Shih Chow
    • 1
  • Henry Teicher
    • 2
  1. 1.Department of StatisticsColumbia UniversityNew YorkUSA
  2. 2.Department of StatisticsRutgers UniversityNew BrunswickUSA

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