Scaling limits of Gaussian vector fields

AbstractFor Gaussian vector fields {X(t) ∈ Rn:t ∈ Rd} we describe the covariance functions of all scaling limits Y(t) = Llimα↓0 B−1(α) X(αt) which can occur when B(α) is a d × d matrix function with B(α) → 0. These matrix covariance functions r(t, s) = EY(t) Y∗(s) are found to be homogeneous in the sense that for some matrix L and each α > 0, (∗) r(αt, αs) = αL∗r(t, s) αL. Processes with stationary increments satisfying (∗) are further analysed and are found to be natural generalizations of Lévy's multiparameter Brownian motion