From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable

We study the Tauberian relations between the moment generating function (MGF) and the complementary cumulative distribution function of a 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.