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Une extension de la loi des grands nombres de Prohorov. (French) Zbl 0535.60011

The Prokhorov strong law of large numbers is extended to random variables taking their values in a 2-uniformly smooth Banach space. The result is as follows:
Let \((X_ n)\) be a sequence of centered, independent r.v. with values in a real, separable, 2-uniformly smooth Banach space \((B,\|\|)\). Suppose that the following hold: i) \(\exists M>0:\forall k\in {\mathbb{N}}\), \(\| X_ k\| \leq(Mk/L_ 2k)\) a.s. (where: \(L_ 2x=\log(\sup(e,\log x)))\); ii) \(\lim_{n\to +\infty}(n^{- 2}\sum_{1\leq k\leq n}E\| X_ k\|^ 2)=0\); iii) \(\forall \epsilon>0\), \(\sum_{n\geq 1}\exp(-\epsilon /\Lambda(n))<+\infty\), where for every integer n: \(\Lambda(n)=2^{-2n}\sum_{2^ n+1\leq k\leq 2^{n+1}}\sup(Ef^ 2(X_ k),\| f\|_ B\leq 1)\). Then: \((1/n)\sum_{1\leq k\leq n}X_ k\to_{n\to +\infty}0\) a.s.

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F15 Strong limit theorems
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References:

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