Application of Tauberian Theorem to the Exponential Decay of the Tail Probability of a Random Variable

Abstract

We give a sufficient condition for the exponential decay of the tail probability of a nonnegative random variable. We consider the Laplace-Stieltjes transform of the probability distribution function of the random variable. We present a theorem, according to which if the abscissa of convergence of the LS transform is negative finite and the real point on the axis of convergence is a pole of the LS transform, then the tail probability decays exponentially. For the proof of the theorem, we extend and apply so-called a finite form of Ikehara's complex Tauberian theorem by Graham-Vaaler.

Tail probability and singularity of Laplace-Stieltjes transform of a Pareto type random variable

“…Similarly to the proofs of various complex Tauberian theorems [5], [6], [7], [10], [11], the main idea for the proof of our Theorem 3.1 is to subtract the singular part from the LS transform ϕ(s). If the singularity is a pole, then the singular part, i.e., the principal part of the pole is a finite sum of rational functions.…”

“…In [9], [10], [11] we studied the asymptotic decay of a light tailed random variable. A random variable X is said to be light tailed if the tail probability P (X > x) decays exponentially, i.e., (1.5) lim x→∞ 1 x log P (X > x) < 0.…”

“…We obtained in [11] the following theorem which gives a sufficient condition for a light tailed random variable.…”

“…Theorem (Nakagawa [11]). For a non-negative random variable X with cumulative distribution function F (x), let ϕ(s) = ∞ 0 e −sx dF (x) be the LaplaceStieltjes transform of F (x) and σ 0 the abscissa of convergence of ϕ(s).…”

“…In the research of the asymptotic decay of a tail probability, we would like to construct a general theory such that a local analytic information on ϕ(s) at s = σ 0 tells the asymptotic evaluation of the tail probability. In a light tailed case [9], [10], [11] we applied to this problem Ikehara's Tauberian theorem [6], [7] and its extension Graham-Vaaler's Tauberian theorem [5], [7]. Ikehara's theorem assumes a global analytic property of ϕ(s), that is, s = σ 0 is a pole and there exist no other singularities on the axis of convergence ℜs = σ 0 .…”