Weak convergence of convex stochastic processes

We discuss the weak convergence of convex stochastic processes. Let {Z(n)(t):t is an element of T}, n greater than or equal to 1, be a sequence of stochastic processes, where T is an open convex set of R-d, such that Z(n):T --> R is a convex function (for each omega and each n). We show that {Z(n)(t):t is an element of T-0} converges weakly to {Z(t):t is an element of T}, for each compact set T-0 of T, if and only if, the finite dimensional distributions of {Z(n)(t):t is an element of T} converge to those of {Z(t):t is an element of T}. This is applied to triangular arrays of empirical processes. In particular, we consider random series and central limit theorems with normal and stable limits. The uniform compact law of the iterated logarithm is also discussed. (C) 1998 Elsevier Science B.V. All rights reserved.