Abstract
Given independent identically distributed random variables {xn;n ∈ ℕq| indexed by q-tuples of positive integers and taking values in a separable Banach space B we approximate the rectangular sums \(\{ \sum\limits_{m \leqq n} {x_m ;n \in \mathbb{N}^q } \} \) by a Brownian sheet. We obtain the corresponding result for random variables with values in a separable Hilbert space H while assuming an optimal moment condition. Generalized versions of the functional law of the iterated logarithm are thus derived.
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