On Small Deviation Probabilities for Some Iterated Random Processes

The asymptotic behavior of the logarithms of small deviation probabilities for some iterated random processes is investigated. It is shown that the logarithmic asymptotics of small deviations of iterated processes depends significantly on the behavior of the distribution function of the iterated process in the neighborhood of its infimum. Some iterated Gaussian processes are considered as examples. In particular, the iterated Slepian process and the iterated fractional Ornstein–Uhlenbeck process are considered.