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Journal of Theoretical Probability

, Volume 13, Issue 1, pp 85–92 | Cite as

Spitzer's Strong Law of Large Numbers in Nonseparable Banach Spaces

  • Berthold Wittje
Article
  • 55 Downloads

Abstract

It is well known, that for the sums of i.i.d. random variables we have Sn/n → 0 a.s. iff ∑n=1 1/nP(|Sn| > ) < ∞ holds for all ε > 0 (Spitzer's SLLN). The result is also known in separable Banach spaces. It will be shown, that this also holds in nonseparable (= not necessarily separable) Banach spaces without any measurability assumption. In the theory of empirical processes this gives a characterization of Glivenko-Cantelli classes.

strong law of large numbers Glivenko–Cantelli class nonmeasurable function 

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© Plenum Publishing Corporation 2000

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  • Berthold Wittje

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