Generalization and refinement of Khintchin's inequality

TL;DR

This paper extends Khintchin's inequality by deriving exponential and power tail bounds for sums of independent random variables, including non-Rademacher types, and explores their limiting behavior.

Contribution

It generalizes Khintchin's inequality to broader classes of variables and norms, providing new tail estimates and limit calculations.

Findings

Derived exponential tail bounds for sums of independent variables.

Extended inequalities to non-Rademacher and non-Lebesgue norms.

Calculated the limits of the tail estimations.

Abstract

We derive the exponential as well as power decreasing tail estimations for normed sums of centered independent identical distributed (or not) random variables on the Khintchine's form. We consider arbitrary, in particular, non-Rademacher's variables and not only Lebesgue-Riesz rearrangement invariant norms for the random variables. We intend to calculate the value of correspondent limit.