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

TL;DR

This paper derives precise exponential tail bounds for sums of Banach space-valued random variables, especially in mixed Lebesgue spaces, enhancing understanding of their probabilistic behavior in the context of the Law of Iterated Logarithm.

Contribution

It provides the first exact exponential tail estimates for sums of Banach space-valued variables in mixed Lebesgue spaces, relevant for PDE analysis.

Findings

Exact exponential tail bounds for Banach space sums

Application to mixed Lebesgue (Bochner) spaces

Examples demonstrating estimate sharpness

Abstract

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.