ON OPERATOR FRACTIONAL LEVY MOTION: INTEGRAL REPRESENTATIONS AND TIME-REVERSIBILITY

In this paper, we construct operator fractional Levy motion (ofLm), a broad class of infinitely divisible stochastic processes that are covariance operator self-similar and have wide-sense stationary increments. The ofLm class generalizes the univariate fractional Levy motion as well as the multivariate operator fractional Brownian motion (ofBm). OfLm can be divided into two types, namely, moving average (maofLm) and real harmonizable (rhofLm), both of which share the covariance structure of ofBm under assumptions. We show that maofLm and rhofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small-and large-scale limiting behavior. We also characterize time-reversibility for ofLm through parametric conditions related to its Levy measure. In particular, we show that, under non-Gaussianity, the parametric conditions for time-reversibility are generally more restrictive than those for the Gaussian case (ofBm).