  It has been recently discovered that some random processes may satisfy limit
theorems even though they exhibit intermittency, namely an unusual growth of
moments. In this paper we provide a deeper understanding of these intricate
limiting phenomena. We show that intermittent processes may exhibit a
multiscale behavior involving growth at different rates. To these rates
correspond different scales. In addition to a dominant scale, intermittent
processes may exhibit secondary scales. The probability of these scales
decreases to zero as a power function of time. For the analysis, we consider
large deviations of the rate of growth of the processes. Our approach is quite
general and covers different possible scenarios with special focus on the
so-called supOU processes.
