Slutsky's theorem states that if random elements u_n converge weakly to random
element u, and v_n converge weakly to constant v (i.e., not random), then
u_n+v_n converge to u+v, and similarly for product and quotient. The usual
proof assumes that the random elements have values in finite dimensional vector
space. However, there are proofs available that work in a metric space. Based
on Van der Vaart's book, Wikipedia discussion and article, and Dudley's paper.
Portmanteau theorem from Billingsley's book.
Please see http://ccm.ucdenver.edu/wiki/Jan_Mandel/Blog for links to the