Self-similarity for information flows with a random load free on distribution: The long memory case

We consider a stochastic model of changing random loads of information flows. The basic random process we exploit is a Double Stochastic Poisson process which manages the change points of the random loads. This Double Stochastic Poisson process is equipped with a Gamma distributed random intensity. The shape parameter of the random intensity equals 2-2H, where 1/2 < H < 1 is the Hurst constant for the corresponding long memory self-similarity. We consider pathwise integral of such kind random load process. Turning to infinity jointly the scale parameter of the random intensity and the variance of the random loads we obtain in a limit a random process with continuous piecewise linear trajectories. Such limit process has the covariance which explicitly coincides with the fractional Brownian motion covariance.