Scaling limits of Gaussian vector fields

For Gaussian vector fields {X(t) [set membership, variant] Rn:t [set membership, variant] Rd} we describe the covariance functions of all scaling limits Y(t) = lim[alpha][downwards arrow]0 B-1([alpha]) X([alpha]t) which can occur when B([alpha]) is a d - d matrix function with B([alpha]) --> 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 [alpha] > 0, (*) r([alpha]t, [alpha]s) = [alpha]L*r(t, s) [alpha]L. Processes with stationary increments satisfying (*) are further analysed and are found to be natural generalizations of Lévy's multiparameter Brownian motion.