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

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  • Berthold Wittje

Abstract

It is well known, that for the sums of i.i.d. random variables we have S n/n → 0 a.s. iff ∑∞ n=1 1/nP(|S n| > nε) 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.

Suggested Citation

  • Berthold Wittje, 2000. "Spitzer's Strong Law of Large Numbers in Nonseparable Banach Spaces," Journal of Theoretical Probability, Springer, vol. 13(1), pages 85-92, January.
    • Handle: RePEc:spr:jotpro:v:13:y:2000:i:1:d:10.1023_a:1007730809136
    DOI: 10.1023/A:1007730809136
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