Date: 12 Sep 2013

Une extension de la loi des grands nombres de Prohorov

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The Prohorov law of large numbers is extended to random variables taking their values in a 2-uniformly smooth Banach space (B, ∥ ∥). In our result the classical assumption of the scalar case is replaced by the following one: $$\forall \varepsilon > 0,\sum\limits_{n \in \mathbb{N}} {\exp } ( - \varepsilon /\Lambda {\text{(}}n{\text{)}}) < + \infty ,$$ where: $$\Lambda {\text{(}}n{\text{)}} = 2^{ - 2n} (\mathop \sum \limits_{2^n + 1 \leqq j \leqq 2^{n + 1} } \sup (Ef^2 (X_j ),\parallel f\parallel _{B'} \leqq 1)).$$