Moments and tails of Lq-valued chaoses based on independent variables with log-concave tails

TL;DR

This paper establishes bounds for the moments of second-order random chaoses with coefficients in Banach spaces, focusing on variables with log-concave tails, and applies these results to Weibull-distributed variables.

Contribution

It provides new lower and upper bounds for moments of Lq-valued chaoses based on independent variables with log-concave tails, extending understanding of their behavior.

Findings

Derived a lower bound for moments of second-order chaoses.

Provided two upper bounds under different assumptions.

Applied bounds to Weibull random variables with shape ≥ 1.

Abstract

We derive a lower bound for moments of random chaoses of order two with coefficients in arbitrary Banach space F generated by independent symmetric random variables with logarithmically concave tails (which is probably two-sided). We also provide two upper bounds for moments of such chaoses when F = L_q. The first is true under the additional subgaussanity assumption. The second one does not require additional assumptions but is not optimal in general. Both upper bounds are sufficient for obtaining two-sided moment estimates for chaoses with values in Lq generated by Weibull random variables with shape parameter greater or equal to 1.