  Precision timing of highly stable milli-second pulsars is a promising
technique for the detection of very low frequency sources of gravitational
waves. In any single pulsar, a stochastic gravitational wave signal appears as
an additional source of timing noise that can be absorbed by the noise model,
and so it is only by considering the coherent response across a network of
pulsars that the signal can be distinguished from other sources of noise. In
the limit where there are many gravitational wave sources in the sky, or many
pulsars in the array, the signals produce a unique tensor correlation pattern
that depends only on the angular separation between each pulsar pair. It is
this distinct fingerprint that is used to search for gravitational waves using
pulsar timing arrays. Here we consider how the prospects for detection are
diminished when the statistical isotropy of the timing array or the
gravitational wave signal is broken by having a finite number of pulsars and a
finite number of sources. We find the standard tensor-correlation analysis to
be remarkably robust, with a mild impact on detectability compared to the
isotropic limit. Only when there are very few sources and very few pulsars does
the standard analysis begin to fail. Having established that the tensor
correlations are a robust signature for detection, we study the use of
"sky-scrambles" to break the correlations as a way to increase confidence in a
detection. This approach is analogous to the use of "time-slides" in the
analysis of data from ground based interferometric detectors.
