ASYMPTOTIC PROPERTIES OF THE FRACTIONAL BROWNIAN-MOTION OF RIEMANN-LIOUVILLE TYPE

We consider the relationship between two types of fractional Brownian motions (fBm) - the Mandelbrot-Van Ness (MV) fBm and the Riemann-Liouville (RL) fBm. It is shown that the covariance of the RL fBm can be split into two parts - a stationary and a nonstationary one. This form of covariance facilitates the study of the asymptotic properties of the RL fBm. In the large-time limit, the RL fBm behaves quite similar to the MV fBm, though not exactly identical, However, increments of the asymptotic fBm are stationary with covariance that is exactly the same as the covariance of increments of the MV fBm. The RL fBm thus exhibits asymptotic fractal properties.