On discretizations of cumulative distribution functions

Stochastic processes in discrete time are considered which develop through the successive application of independent positive multipliers and also are martingales. We construct optimal discretizations and derive properties of the Mellin-Stieltjes transforms of the cumulative distribution functions of the multipliers. Discretization means approximation by positive random variables with values in a given discrete set. It will be shown that the independence of the factors will be preserved in this procedure. The important case that discretization leads to multipliers with values in some fixed geometric progression allows one to write the Mellin-Stieltjes transforms as Laurent series. The processes are then investigated by using the fact that the Mellin-Stieltjes transform of an independent product is the product of the transforms of its factors. (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim