  We obtain quenched hitting distributions to be compound Poissonian for a
certain class of random dynamical systems. The theory is general and designed
to accommodate non-uniformly expanding behavior and targets that do not overlap
much with the region where uniformity breaks. Based on annealed and quenched
polynomial decay of correlations, our quenched result adopts annealed Kac-type
time-normalization and finds limits to be noise-independent. The technique
involves a probabilistic blockapproximation where the quenched hit-counting
function up to annealed Kac-normalized time is split into equally sized blocks
which are mimicked by an independency of random variables distributed just like
each of them. The theory is made operational due to a result that allows
certain hitting quantities to be recovered from return quantities. Our
application is to a class of random piecewise expanding one-dimensional
systems, casting new light on the well-known deterministic dichotomy between
periodic and aperiodic points, their usual extremal index formula
EI=1-1/JT^p(x_0) , and recovering the PolyaAeppli case for general
Bernoulli-driven systems, but distinct behavior otherwise. Future and on-going
investigations aim to produce and accommodate examples of bonafide nonuniformly
expanding random systems and targets approaching their neutral points.
