On the supremum of certain families of stochastic processes

We consider a family of stochastic processes {X-t(epsilon), t is an element of T} on a metric space T, with a parameter epsilon down arrow 0. We study the conditions under which lim(epsilon -> 0)(P)(sup(r is an element of T)vertical bar X-t(epsilon)vertical bar < delta) = 1 In our main result in Section 2, we find conditions in terms of the covering number N(T, d, delta) that ensure (1.1) holds. Although our technique is based on well known chaining methods, our principal result appears to be new. In Section 3 we discuss briefly the optimality of our hypotheses and compare our theorem with the Kolmogorov criterion for continuity of stochastic processes. In Section 4 we present an application of our main theorem to random fields constructed from Levy random measures.