Return times at periodic points in random dynamics

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.