On the Laplace Transform of the Lognormal Distribution

Abstract

Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. A wide variety of methods have been employed to provide approximations, both analytical and numerical. In this paper, we analyze a closed-form approximation L(θ) of the Laplace transform L(θ) which is obtained via a modified version of Laplace's method. This approximation, given in terms of the Lambert W (•) function, is tractable enough for applications. We prove that L(θ) is asymptotically equivalent to L(θ) as θ → ∞. We apply this result to construct a reliable Monte Carlo estimator of L(θ) and prove it to be logarithmically efficient in the rare event sense as θ → ∞.

Unified Importance Sampling Schemes for Efficient Simulation of Outage Capacity Over Generalized Fading Channels

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