Integrability of Freely Infinitely Divisible Distributions and L\'evy Measures

Abstract

Under a growth condition on an increasing function $g$, we prove that integrability of a freely infinitely divisible distribution with respect to $g$ is equivalent to that of the large-jump part of its free L\'evy measure. For every increasing freely submultiplicative function $g$, integrability of the distribution implies integrability of the large-jump part of its free L\'evy measure. We also obtain two-sided tail comparisons between freely infinitely divisible distributions and their free L\'evy measures, uniform integrability along free convolution semigroups, and integrability results and tail estimates for fractional free convolution powers.