  Pulsar timing arrays (PTA) hunt for gravitational waves (GW) by searching for
the correlations that GWs induce in the time-of-arrival residuals from
different pulsars. If the GW sources are of astrophysical origin, then they are
located at discrete points on the sky. However, PTA data are often modeled, and
subsequently analyzed, via a "standard Gaussian ensemble". That ensemble is
obtained in the limit of an infinite density of vanishingly weak,
Poisson-distributed sources. In this paper, we move away from that ensemble, to
study the effects of two types of "source anisotropy". The first (a), which is
often called "shot noise", arises because there are $N$ discrete GW sources at
specific sky locations. The second (b) arises because the GW source positions
are not a Poisson process, for example, because galaxy locations are clustered.
Here, we quantify the impact of (a) and (b) on the mean and variance of the
pulsar-averaged Hellings and Downs correlation. For conventional PTA sources,
we show that the effects of shot noise (a) are much larger than the effects of
clustering (b).
