A note on recovering the distributions from exponential moments

The problem of recovering a cumulative distribution function of a positive random variable via the scaled Laplace transform inversion is studied. The uniform upper bound of proposed approximation is derived. The approximation of a compound Poisson distribution as well as the estimation of a distribution function of the summands given the sample from a compound Poisson distribution are investigated. Applying the simulation study, the question of selecting the optimal scaling parameter of the proposed Laplace transform inversion is considered. The behavior of the approximants are demonstrated via plots and table.