CONVERGENCE IN DISTRIBUTION OF CONDITIONAL EXPECTATIONS
成果类型：
Article
署名作者：
GOGGIN, EM
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/aop/1176988743
发表日期：
1994
页码：
1097-1114
关键词：
摘要：
Suppose the random variables (X(N), Y(N)) on the probability space (OMEGA(N), F(N), P(N)) converge in distribution to the pair (X, Y) on (OMEGA, F, P), as N --> infinity. This paper seeks conditions which imply convergence in distribution of the conditional expectations E(PN){F(X(N))\Y(N)} to E(P){F(X)\Y}, for all bounded continuous functions F. An absolutely continuous change of probability measure is made from P(N) to a measure Q(N) under which X(N) and Y(N) are independent. The Radon-Nikodym derivative dP(N)/dQ(N) is denoted by L(N). Similarly, an absolutely continuous change of measure from P to Q is made, with Radon-Nikodym derivative dP/dQ = L. If the Q(N)-distribution of (X(N), Y(N), L(N)) converges weakly to the Q-distribution of (X, Y, L), convergence in distribution of E(PN)(F(X(N)\Y(N)) (under the original distributions) to E(P){F(X)\Y} follows. Conditions of a uniform equicontinuity nature on the L(N) are presented which imply the required convergence. Finally, an example is given, where convergence of the conditional expectations can be shown quite easily.