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)


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.


Banach Space Independent Random Variable Similar Spirit Infinite Dimensional Hilbert Space Kolmogorov Type 
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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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