Analytic continuations of Fourier and Stieltjes transforms and generalized moments of probability measures

TL;DR

This paper introduces a method to extend the concept of moments for probability measures via analytic continuations of Fourier and Stieltjes transforms, unifying different generalizations and applying them to convergence proofs.

Contribution

It proposes a unified approach to defining complex moments through analytic continuations, connecting Fourier and Stieltjes transform-based generalizations.

Findings

Unified complex moments for probability measures.

Short proofs of convergence to Cauchy distributions.

Applicability to various convolution types.

Abstract

We consider analytic continuations of Fourier transforms and Stieltjes transforms. This enables us to define what we call complex moments for some class of probability measures which do not have moments in the usual sense. There are two ways to generalize moments accordingly to Fourier and Stieltjes transforms; however these two turn out to coincide. As applications, we give short proofs of the convergence of probability measures to Cauchy distributions with respect to tensor, free, Boolean and monotone convolutions.