STOCHASTIC MONOTONICITY AND CONDITIONING IN THE LIMIT

Suppose that {(X-n, Y-n)} is a sequence of pairs of vector-valued stoc hastic variables which converges weakly to (X, Y), and that {y(n)} con verges to y. Sufficient conditions for the conditional distribution of X-n, given Y-n = y(n), to converge to the conditional distribution of X given Y = y are given in terms of stochastic monotonicity. Conditio ns, which guarantee that also moments of the conditional distributions converge to the moments of the ones of the limit, are also derived.