From moment explosion to the asymptotic behavior of the cumulative distribution for a random variable

TL;DR

This paper investigates the relationship between the moment generating function and the tail behavior of a distribution, especially near critical moments, with applications to CIR processes and their integrals.

Contribution

It establishes Tauberian relations linking the MGF's behavior near critical points to the distribution's tail, extending understanding of tail asymptotics for complex stochastic models.

Findings

Derived relations between MGF and tail behavior near critical moments

Applied results to superpositions of CIR processes and their integrals

Provided insights into the asymptotic tail behavior of complex stochastic models

Abstract

We study the Tauberian relations between the moment generating function (MGF) and the complementary cumulative distribution function of a random variable whose MGF is finite only on part of the real line. We relate the right tail behavior of the cumulative distribution function of such a random variable to the behavior of its MGF near the critical moment. We apply our results to an arbitrary superposition of a CIR process and the time-integral of this process.