An operational calculus for probability distributions via Laplace transforms

In this paper the authors investigate operators that map one or more probability distributions on the positive real line into another via their Laplace-Stieltjes transforms. Their goal is to make it easier to construct new transforms by manipulating known transforms. The authors envision the results here assisting modelling in conjunction with numerical transform inversion software. They primarily focus on operators related to infinitely divisible distributions and Lévy processes, drawing upon Feller. The authors give many concrete examples of infinitely divisible distributions. They consider a cumulant-moment-transfer operator that allows us to relate the cumulants of one distribution to the moments of another. The authors consider a power-mixture operator corresponding to an independently stopped Lévy process. The special case of exponential power mixtures is a continuous analogue of geometric random sums. The authors introduce a further special case which is remarkably tractable, exponential mixtures of inverse Gaussian distributions (EMIGs). EMIGs arise naturally as approximations for busy periods in queues. They show that the staday-state waiting time in an M/G/1 queue is the difference of two EMIGs when the service-time distribution is an EMIG. The authors consider several transforms related to first-passage times, e.g. for te M/M/1 queue, reflected Brownian motion and Lévy processes. Some of the associated probability density functions involve Bessel functions and theta functions. The authors describe properties of the operators, including how they transform moments.