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Some Generalized Martingales Arising from the Strong Law of Large Numbers

  • Bernard Heinkel
Conference paper
Part of the Progress in Probability book series (PRPR, volume 35)

Abstract

Let (X k) be a sequence of independent random variables, defined on a probability space (Ω,.F, P) and taking their values in a real, separable, Banach space (B, || ||) (that Banach space being equipped with its Borel σ — field B ). To that sequence (X k) one associates the partial sums : S n = X 1 + … + X n and the σ- fields F n generated by X 1,…, X n. One says that (X k) satisfies the weak law of large numbers (WLLN) if ( ||S n/n|| ) converges to 0 in probability; the strong law of large numbers (SLLN) holds for (X k) if (||S n/n||) converges a.s. to 0.

Keywords

Banach Space Independent Random Variable Similar Spirit Infinite Dimensional Hilbert Space Kolmogorov Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Bernard Heinkel
    • 1
  1. 1.Département de MathématiqueUniversité Louis PasteurStrasbourgFrance

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