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-nℱ-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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
D. G. Austin, “A sample property of martingales,”Ann. Math. Stat.37 (1966), 1396–1397.
D. Blackwell, “On optimal systems,”Ann. Math. Stat.25 (1954), 394–397.
B. M. Brown, “A note on convergence of moments,”Ann. Math. Stat.42 (1971), 777–779.
D. L. Burkholder, “Martingale transforms,”Ann. Math. Stat.37 (1966), 1494–1504.
D. L. Burkholder, “Distribution function inequalities for martingales,”Ann. Probability1 (1973), 19–42.
D. L. Burkholder and R. F. Gundy, “Extrapolation and interpolation of quasi-linear operators on martingales,”Acta Math.124 (1970), 249–304.
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.
Y.S. Chow, “On a strong law of large numbers for martingales,”Ann. Math. Stat.38 (1967), 610.
Y. S. Chow, “Convergence of sums of squares of martingale differences,”Ann. Math. Stat.39 (1968), 123–133.
Y. S. Chow, “On the Le-convergence forn - 1 /PS„0<p <2,“Ann. Math. Stat.42 (1971), 393–394.
K. L. ChungA Course in Probability TheoryHarcourt Brace, New York, 1968; 2nd ed., Academic Press, New York, 1974.
B. Davis, “A comparison test for martingale inequalities,”Ann. Math. Stat.40 (1969), 505–508.
J. L. DoobStochastic ProcessesWiley, New York, 1953.
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.
L. E. Dubins and L. J. SavageHow to Gamble If You MustMcGraw-Hill, New York, 1965.
A. M. Garsia, “On a convex function inequality for martingales,”Ann. Probability1 (1973), 171–174.
R. F. Gundy, “A decomposition for 2’I-bounded martingales,”Ann. Math. Stat.39 (1968), 134–138.
J. NeveuMartingales à temps discretsMasson, Paris, 1972.
R. Panzone, “Alternative proofs for certain uperossing inequalities,”Ann. Math. Stat.38 (1967), 735–741.
E. M. SteinTopics in Harmonic Analysis Related to the Littlewood—Paley TheoryPrinceton Univ. Press, Princeton, 1970.
H. Teicher, “Moments of randomly stopped sums-revisited,”Jour. Theor. Prob.9 (1995), 779–793.
A. ZygmundTrigonometric SeriesVol. I, Cambridge, 1959.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media New York
About this chapter
Cite this chapter
Chow, Y.S., Teicher, H. (1997). Martingales. In: Probability Theory. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1950-7_11
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1950-7_11
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-40607-7
Online ISBN: 978-1-4612-1950-7
eBook Packages: Springer Book Archive