AN OPERATIONAL CALCULUS FOR PROBABILITY-DISTRIBUTIONS VIA LAPLACE TRANSFORMS

In this paper we investigate operators that mal, one or more probabili ty distributions on the positive real line into another via their Lapl ace-Stieltjes transforms. Our goal is to make it easier to construct n ew transforms by manipulating known transforms. We envision the result s here assisting modelling in conjunction with numerical transform inv ersion software. We primarily focus on operators related to infinitely divisible distributions and Levy processes, drawing upon Feller (1971 ). We give many concrete examples of infinitely divisible distribution s. We consider a cumulant-moment-transfer operator that allows us to r elate the cumulants of one distribution to the moments of another. We consider a power-mixture operator corresponding to an independently st opped Levy process. The special case of exponential power mixtures is a continuous analog of geometric random sums. We introduce a further s pecial case which is remarkably tractable, exponential mixtures of inv erse Gaussian distributions (EMIGs). EMIGs arise naturally as approxim ations for busy periods in queues. We show that the steady-state waiti ng time in an M/G/1 queue is the difference of two EMIGs when the serv ice-time distribution is an EMIG. We consider several transforms relat ed to first-passage times, e.g. for the M/M/1 queue, reflected Brownia n motion and Levy processes. Some of the associated probability densit y functions involve Bessel functions and theta functions. We describe properties of the operators, including how they transform moments.