Comments on the continuity of distribution functions obtained by superposition.

Abstract

Assuming that Y is a nonnegative random variable independent of the differential process X(t), attention is given to the question of whether or not the superposition X(Y) can have a continuous probability distribution. If the process has continuous distributions, then the superposition is continuous if and only if P/Y = 0/ = 0. If the process has discontinuous distributions and no trend, then no superposition can have continuous distribution. If the process has discontinuous distributions and nonzero trend, then the superposition onto a random epoch has continuous distribution if and only if Y has continuous distribution.