Spitzer's Strong Law of Large Numbers in Nonseparable Banach Spaces
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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| > nε) < ∞ 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.
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