Weak convergence of convex stochastic processes

We discuss the weak convergence of convex stochastic processes. Let {Zn(t):t [set membership, variant] T}, n [greater-or-equal, slanted] 1, be a sequence of stochastic processes, where T is an open convex set of , such that is a convex function (for each [omega] and each n). We show that {Zn(t):t [set membership, variant] T0} converges weakly to {Z(t):t [set membership, variant] T}, for each compact set T0 of T, if and only if, the finite dimensional distributions of {Zn(t):t [set membership, variant] T} converge to those of {Z(t):t [set membership, variant] 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.