We consider a sequence of i.i.d. random variables, (Formula presented.), (Formula presented.), (Formula presented.), and subordinate it by a doubly stochastic Poisson process (Formula presented.), where (Formula presented.) is a random variable and (Formula presented.) is a standard Poisson process. The subordinated continuous time process (Formula presented.) is known as the PSI-process. Elements of the triplet (Formula presented.) are supposed to be independent. For sums of n, independent copies of such processes, normalized by (Formula presented.), we establish a functional limit theorem in the Skorokhod space (Formula presented.), for any (Formula presented.), under the assumption (Formula presented.) for some (Formula presented.). Here, (Formula presented.) reflects the tail behavior of the distribution of (Formula presented.), in particular, (Formula presented.) when (Formula presented.). The limit process is a stationary Gaussian process with the covariance function (Formula presented.), (Formula presented.). As a sample application, we construct a martingale from the PSI-process and establish a convergence of normalized cumulative sums of such i.i.d. martingales.