The purpose of this paper is twofold. From the viewpoint of communications engineering, it presents a model of certain random perturbations that appear to come in clusters, or bursts. This will be achieved by introducing the concept of “self-similar stochastic point process in continuous time.” The resulting mechanism presents fascinating peculiarities from the mathematical viewpoint. In order to make them more palatable as well as to help in the search for further developments, the basic concept of “conditional stationarity” will be discussed in greater detail than would be strictly necessary from the viewpoint of the immediate engineering problem of errors of transmission. This work is related to an earlier “new model” due to J. M. Berger and the author; the logical structure of the theory has been further streamlined and a number of fresh consequences have been derived; the empirical fit has been further improved, while recourse to ad hoc corrective terms was made unnecessary. Copyright © 1965 by The Institute of Electrical and Electronics Engineers, Inc.