  We prove a quenched limiting law for random measures on subshifts at periodic
points. We consider a family of measures $\{\mu_\omega\}_{\omega\in\Omega}$,
where the `driving space' $\Omega$ is equipped with a probability measure which
is invariant under a transformation $\theta$. We assume that the fibred
measures $\mu_\omega$ satisfy a generalised invariance property and are
$\psi$-mixing. We then show that for almost every $\omega$ the return times to
cylinders $A_n$ at periodic points are in the limit compound Poisson
distributed for a parameter $\vartheta$ which is given by the escape rate at
the periodic point.
