ABSTRACT
Associated with a Po´lya urn growing in discrete time is a process obtained by embedding it in continuous time. It comprises a (changing) number of parallel Poisson processes. Poissonization has been used as a heuristic to understand discrete processes for some time. Early work on poissonization can be traced back to Kac (1949). In the context of Po´lya urns, the embedding was introduced in Athreya and Karlin (1968) to model the growth of an urn in discrete time according to certain rules. Although it has been around for some time, no one until very recently (see Balaji and Mahmoud (2006)) had called it by the name the Po´lya process. We think it is an appropriate name. We shall refer to the embedded process by that name. Of course, a Po´lya urn growing in discrete time defines a stochastic process, too. However, to create a desirable distinction, we shall refer to a discrete Po´lya system as we did before as a scheme, and reserve the title Po´lya process to the poissonized process obtained by embedding the scheme in continuous time.
