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.
Notes
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