An infinite-dimensional law of large numbers in Cesaro's sense (Q919698)

The author proves the following theorem: Let the random variable X take its values in a real, separable Banach space \((B,\| \cdot \|)\), equipped with its Borel \(\sigma\)-field, and \(\{X_ k\}\) be a sequence of independent copies of X. Then the statements (i) for all \(\alpha\in (0,1)\) the sequence \(V_ n=(1/A_ n^{\alpha})\sum^{n}_{k=0}A_{n-k}^{\alpha -1}X_ k\) converges a.s. to E(X), where \[ A_ 0^{\beta}=1,\quad A_ 1^{\beta}=\beta +1,...,\quad A_ k^{\beta}=(\beta +1)...(\beta +k)/k!,...\text{ for } every\quad real\quad \beta >-1. \] (ii) \(\| X\|^{1/\alpha}\) is integrable. are equivalent.