  There is growing interest in the overlap reduction function in pulsar timing
array observations as a probe of modified gravity. However, current
approximations to the Hellings-Downs curve for subluminal gravitational wave
propagation, say $v<1$, diverge at small angular pulsar separation. In this
paper, we find that the ORF for the $v<1$ case is sensitive to finite distance
effects. First, we show that finite distance effects introduce an effective
cut-off in the spherical harmonics decomposition at $\ell\sim \sqrt{1-v^2} \,
kL$, where $\ell$ is the multipole number, $k$ the wavenumber of the
gravitational wave and $L$ the distance to the pulsars. Then, we find that the
overlap reduction function in the small angle limit approaches a value given by
$\pi kL\,v^2\,(1-v^2)^2$ times a normalization factor, exactly matching the
value for the autocorrelation recently derived. Although we focus on the $v<1$
case, our formulation is valid for any value of $v$.
