Tightness of the sums of independent identically distributed pseudo-poissonian processes in the Skorokhod space

We consider pseudo-poissonian process of the following simple type: it is a poissonian subordinator for a sequence of i.i.d. random variables with a finite variance. Next we consider sums of i.i.d. copies of such pseudo-poissonian process. For a family of the distributions of these random sums we prove the tightness (relative compactness) in the Skorokhod space. Under conditions of the Central Limit Theorem for vectors we obtain a weak convergence in the functional Skorokhod space of the examined sums to the Ornstein–Uhlenbeck process.