  We give a sufficient condition for the exponential decay of the tail
probability of a non-negative 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.
