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


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


Nondecreasing Function Martingale Difference Uniform Integrability Nonnegative Measurable Function Stochastic Sequence 
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  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. Probability 1 (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, 610.Google 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 L-convergence for L p-convergence for n -1/p S n, 0< p < 2,” Ann. Math. Stat. 42 (1971), 393–394.zbMATHCrossRefGoogle Scholar
  11. K. L. Chung, A Course in Probability Theory, Harcourt Brace, New York, 1968; 2nd ed., Academic Press, New York, 1974.zbMATHGoogle Scholar
  12. B. Davis, “A comparison test for martingale inequalities,” Ann. Math. Stat. 40 (1969), 505–508.zbMATHCrossRefGoogle Scholar
  13. J. L. Doob, Stochastic Processes, Wiley, 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. Savage, How to Gamble If You Must, McGraw-Hill, New York, 1965.zbMATHGoogle Scholar
  16. A. M. Garsia, “On a convex function inequality for martingales,” Ann. Probability 1 (1973), 171–174.MathSciNetzbMATHCrossRefGoogle Scholar
  17. R. F. Gundy, “A decomposition for if rbounded martingales,” Ann. Math. Stat. 39 (1968), 134–138.MathSciNetzbMATHCrossRefGoogle Scholar
  18. J. Neveu, Martingales à temps discrets, Masson, Paris, 1972.Google Scholar
  19. R. Panzone, “Alternative proofs for certain upcrossing inequalities,” Ann. Math. Stat. 38 (1967), 735–741.MathSciNetzbMATHCrossRefGoogle Scholar
  20. E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton Univ. Press, Princeton, 1970.zbMATHGoogle Scholar
  21. A. Zygmund, Trigonometric Series, Vol. I, Cambridge, 1959.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

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

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