An infinite-dimensional law of large numbers in Cesaro's sense

Let (X _k) _k≥0 be a sequence of independent copies of a random variableX taking its values in a real separable Banach space (B, ¦ ¦). For every real number β>?1 one defines the following coefficients: $$A_0^\beta = 1, A_1^\beta = \beta + 1,..., A_k^\beta = (\beta + 1) \cdots (\beta + k)/k!,...$$ It is shown that for all α∈]0, 1 the sequenceV _n =(1/A _n ^α )∑_0?k?n A _n?k ^α?1 X _k converges almost surely toE(X) if and only if ‖X‖^1/α is integrable. This extends results obtained earlier by several authors for scalar-valued random variables: Lorentz (case 1/2<α<1), Chow and Lai (case 0<α<1/2), Déniel and Derriennic (case α=1/2).