Limit theorems for small deviation probabilities of some iterated stochastic processes

We derive logarithmic asymptotics for probabilities of small deviations for some iterated processes. We show that under appropriate conditions, these asymptotics are the same as those of the processes which generate the iterated processes. When these conditions do not hold, the asymptotics of small deviations for iterated processes are quite different. We apply our results to the iterated processes generated by the compound Cox processes and the compound renewal processes.