Cesàro convergence of martingale difference sequences and the Banach-Saks and Szlenk theorems
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- by Francisco J. Freniche PDF
- Proc. Amer. Math. Soc. 103 (1988), 234-236 Request permission
Abstract:
It is shown that, for $1 \leq p < + \infty$, any weakly null martingale difference sequence in ${L^p}\left [ {0,1} \right ]$ is Cesàro convergent to zero in the ${L^p}$ norm. This result combined with a theorem of Gaposhkin gives an easy proof of two theorems of Banach-Saks and Szlenk at once.References
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S. Banach and S. Saks, Sur la convergence forte dans les champs ${L^p}$, Studia Math. 2 (1930), 51-57.
V. F. Gaposhkin, Convergence and limits theorems for sequences of random variables, Theor. Probab. Appl. 17 (1972), 379-399.
- W. Szlenk, Sur les suites faiblement convergentes dans l’espace $L$, Studia Math. 25 (1965), 337–341 (French). MR 201956, DOI 10.4064/sm-25-3-337-341
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 234-236
- MSC: Primary 60F25; Secondary 46E30, 60G42
- DOI: https://doi.org/10.1090/S0002-9939-1988-0938674-3
- MathSciNet review: 938674