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

Return times at periodic points in random dynamics