  Detection of a gravitational-wave stochastic background via ground or
space-based gravitational-wave detectors requires the cross-correlation of the
response of two or more independent detectors. The cross-correlation involves a
frequency-dependent factor -- the so-called overlap reduction function or
Hellings-Downs curve -- that depends on the relative geometry of each detector
pair: i.e., the detector separations and the relative orientation of their
antenna patterns (beams). An incorrect formulation of this geometrical factor
has appeared in the literature, leading to incorrect conclusions regarding the
sensitivity of proposed detectors to a stochastic gravitational-wave
background. To rectify these errors and as a reference for future work we
provide here a complete, first-principles derivation of the overlap reduction
function and assess the nature of the errors associated with the use of the
incorrect expression that has appeared in the literature. We describe the
behavior of the overlap reduction function in different limiting regimes, and
show how the difference between the correct and incorrect expressions can be
understood physically.
