Tail probability of random variable and Laplace transform

Abstract

We investigate the exponential decay of the tail probability P(X > x) of a continuous type random variable X. Let ϕ(s) be the Laplace–Stieltjes transform of the probability distribution function F(x) = P(X ≤ x) of X, and σ₀ be the abscissa of convergence of ϕ(s). We will prove that if −∞ < σ₀ < 0 and the singularities of ϕ(s) on the axis of convergence are only a finite number of poles, then the tail probability decays exponentially. For the proof of our theorem, Ikehara's Tauberian theorem will be extended and applied.

Keywords:

• Tail probability of random variable
• Exponential decay
• Laplace transform
• Tauberian theorem
• Mathematics Subject Classifications (2000): Primary 44A10
• 60A99

Notes

Emaill: [email protected]

Additional information

Notes on contributors

Kenji Nakagawa *

Emaill: [email protected]