In this paper, processes in R d of the form X(t)=(X 1 (t), X 2 (t), ⋯, X N (t), where X i (t) is a stable process of index α i in Euclidean space of dimension d i and d=d 1 + ⋯ + d N , are considered. The asymptotic behaviour of the first passage time out of a sphere, and of the sojourn time in a sphere is established. Properties of the space-time process (X 1 (t), t) in R d+1 are obtained when X 1 (t) is a stable process in R d . For each of these processes, a Hausdorff measure function θ(h) is found such that the range set R(s) of the sample path on [0, s] has Hausdorff θ-measure c s for a suitable finite positive c.