---
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.\
    AMS Subject Classification: 60B10; 30D20; 46L53; 46L54\
    Keywords: Fourier transform; Stieltjes transform; Cauchy distribution; non-commutative probability theory; Paley-Wiener theorem
author:
- |
    Takahiro Hasebe\
    Graduate School of Science, Kyoto University,\
    Kyoto 606-8502, Japan\
    E-mail: hsb@kurims.kyoto-u.ac.jp
title: Analytic continuations of Fourier and Stieltjes transforms and generalized moments of probability measures
---

Observation on Cauchy distributions {#Cauchy}
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Let $\mu_{a,b}$ be the Cauchy distribution $$\mu_{a,b}(dx) = \frac{1}{\pi} \cdot \frac{b}{(x-a)^2 + b^2} dx$$ with parameters $a \in \operatorname*{\mathbb{R}}$ and $b > 0$. This distribution has many important properties in probability theory, which we call tensor probability theory, and also in noncommutative probability theory, especially in free, Boolean and monotone probability theories [@Mur3; @S-W; @V2]. For instance, $\mu_{a,b}$ is an infinitely divisible 