We obtain the quite exact exponential bounds for tails of distributions of
sums of Banach space valued random variables uniformly over the number of
summands under natural for the Law of Iterated Logarithm (LIL) norming.
We study especially the case of the so-called mixed (anisotropic)
Lebesgue-Riesz spaces, on the other words, Bochner's spaces, for instance,
continuous-Lebesgue spaces, which appear for example in the investigation of
non-linear Partial Differential Equations of evolutionary type.
We give also some examples in order to show the exactness of our estimates.
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