Exact exponential tail estimations in the Law of Iterated Logarithm for Bochner's mixed Lebesgue spaces

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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.