Sums of Independent Pseudopoissonian Processes and Their Convergence to the $\alpha$-stable and Fractional Ornstein-Uhlenbeck-Levy Processes

The Pseudopoissonian processes are studied in chapter X (v.2) of the famous Feller's monograph. They are introduced in a sense of subordination of Markov sequence to the independent "leading" Poisson process. We consider sums of such independent Pseudopoisson processes for which the subordinate sequence consists of i.i.d. random variables belonging to the attraction domain of the symmetrical $\alpha$-stable law including the gaussian case. Under certain distributions of the total intensities of the "leading" Poisson processes we obtain as a limit the processes which are described in models for Telecom processes (I.Kaj, M.S.Taqqu, R.L.Wolpert), and which are named (by authors) as $\alpha$-stable, fractional Ornstein-Uhlenbeck-Levy processes. In talk we discuss applications of the proposed scheme to a stochastic modelling of the interest rates.