  We study the number of white balls in a classical P\'olya urn model with the
additional feature that, at random times, a black ball is added to the urn. The
number of draws between these random times are i.i.d. and, under certain moment
conditions on the inter-arrival distribution, we characterize the limiting
distribution of the (properly scaled) number of white balls as the number of
draws goes to infinity. The possible limiting distributions obtained in this
way vary considerably depending on the inter-arrival distribution and are
difficult to describe explicitly. However, we show that the limits are fixed
points of certain probabilistic distributional transformations, and this fact
provides a proof of convergence and leads to properties of the limits. The
model can alternatively be viewed as a preferential attachment random graph
model where added vertices initially have a random number of edges, and from
this perspective, our results describe the limit of the degree of a fixed
vertex.
