Some definite integrals arising from selfdecomposable characteristic functions

TL;DR

This paper explores definite integrals derived from selfdecomposable characteristic functions, revealing new formulas through their integral representations related to Lévy processes.

Contribution

It introduces new definite integral formulas based on the properties of selfdecomposable distributions and their integral representations with respect to Lévy processes.

Findings

Derived new formulas for definite integrals

Connected characteristic functions to integral representations

Potentially discovered previously unknown integrals

Abstract

In the probability theory \emph{selfdecomposable, or class $L_0$ distributions} play an important role as they are limiting distributions of normalized partial sums of sequences of independent, not necessarily identically distributed, random variables. The class $L_0$ is quite large and includes many known classical distributions and statistics. For this note the most important feature of the selfdecomposable variables are their random integral representation with respect to L\'evy process. From those random integral representation we get equality of logarithms of some characteristic functions. These allows us to get formulas for some definite integrals, some of them probably were unknown before.