Martingale convergence via the square function
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- by Louis H. Y. Chen PDF
- Proc. Amer. Math. Soc. 83 (1981), 125-127 Request permission
Abstract:
By exploiting the natural setting of a convergence theorem of Burkholder, a direct and elementary proof of the theorem is given. This proof is also new for the martingale convergence theorem and the martingale transform convergence theorem which are corollaries to the above-mentioned convergence theorem.References
- D. G. Austin, A sample function property of martingales, Ann. Math. Statist. 37 (1966), 1396–1397. MR 198532, DOI 10.1214/aoms/1177699283
- Luis Báez-Duarte, Another look at the martingale theorem, J. Math. Anal. Appl. 23 (1968), 551–557. MR 229285, DOI 10.1016/0022-247X(68)90136-4
- Luis Báez-Duarte, On the convergence of martingale transforms, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 19 (1971), 319–322. MR 300328, DOI 10.1007/BF00535836
- D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504. MR 208647, DOI 10.1214/aoms/1177699141
- D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probability 1 (1973), 19–42. MR 365692, DOI 10.1214/aop/1176997023
- Louis H. Y. Chen, A martingale inequality for the square and maximal functions, Ann. Probab. 7 (1979), no. 6, 1051–1055. MR 548899
- Burgess Davis, On the integrability of the martingale square function, Israel J. Math. 8 (1970), 187–190. MR 268966, DOI 10.1007/BF02771313
- Adriano M. Garsia, Martingale inequalities: Seminar notes on recent progress, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. MR 0448538
- Richard Isaac, A proof of the martingale convergence theorem, Proc. Amer. Math. Soc. 16 (1965), 842–844. MR 179830, DOI 10.1090/S0002-9939-1965-0179830-9
- Charles W. Lamb, A short proof of the martingale convergence theorem, Proc. Amer. Math. Soc. 38 (1973), 215–217. MR 324770, DOI 10.1090/S0002-9939-1973-0324770-X
- Paul-André Meyer, Martingales and stochastic integrals. I. Lecture Notes in Mathematics. Vol. 284, Springer-Verlag, Berlin-New York, 1972. MR 0426145
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 83 (1981), 125-127
- MSC: Primary 60G42
- DOI: https://doi.org/10.1090/S0002-9939-1981-0619996-6
- MathSciNet review: 619996