Weconsider a PSI-process, that is a sequence of random variables (ζ_i), i = 0.1, ..., which is subordinated by a continuous-time non-decreasing integer-valued process N(t): φ(t) = ζ_N(t). Our main example is when N(t) itself is obtained as a subordination of the standard Poisson process Π(s) by a non-decreasing Lévy process S(t): N(t) = Π(S(t)).The values (ζ_i)one interprets as random claims, while their accumulated intensity S(t) is itself random. We show that in this case the process N(t) is a compound Poisson process of the stuttering type and its rate depends just on the value of theLaplace exponent of S(t) at 1. Under the assumption that the driven sequence (ζ_i) consists of i.i.d. random variables with finite variance we calculate a correlation function of the constructed PSI-process. Finally, we show that properly rescaled sums of such processes converge to the Ornstein-Uhlenbeck process in the Skorokhod space. We suppose that the results stated in the paper mightbe interesting for theorists and practitioners in insurance, in particular, for solution of reinsurance tasks.