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Cesàro convergence of martingale difference sequences and the Banach-Saks and Szlenk theorems


Author: Francisco J. Freniche
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
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [BS] S. Banach and S. Saks, Sur la convergence forte dans les champs $ {L^p}$, Studia Math. 2 (1930), 51-57.
  • [G] V. F. Gaposhkin, Convergence and limits theorems for sequences of random variables, Theor. Probab. Appl. 17 (1972), 379-399.
  • [S] W. Szlenk, Sur les suites faiblement convergentes dans l’espace 𝐿, Studia Math. 25 (1965), 337–341 (French). MR 0201956

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0938674-3
Keywords: Martingale difference sequence, Cesàro convergence in the $ {L^p}$ norm, weakly null sequences in $ {L^p}$
Article copyright: © Copyright 1988 American Mathematical Society