Stochastic Monotonicity and Conditioning in the Limit

• Published in: Scandinavian journal of statistics Vol. 25; no. 3; pp. 569 - 572
• Main Author: Nerman, Olle
• Format: Journal Article
• Language: English
• Published: Oxford, UK and Boston, USA Blackwell Publishers Ltd 01.09.1998; Blackwell Publishers
• Subjects: Conditional convergence; conditional distribution; Distribution functions; Gaussian distributions; Interior points; Logical givens; Mathematical moments; Mathematical theorems; Perceptron convergence procedure; Rectangles; Statistical theories; stochastic monotonicity; weak convergence
• ISSN: 0303-6898; 1467-9469
• DOI: 10.1111/1467-9469.00121

Suppose that$\{(X_{n},Y_{n})\}$is a sequence of pairs of vector-valued stochastic variables which converges weakly to (X, Y), and that$\{y_{n}\}$converges to y. Sufficient conditions for the conditional distribution of Xngiven$Y_{n}=y_{n}$to converge to the conditional distribution of X given Y = y are given in terms of stochastic monotonicity. Conditions, which guarantee that also moments of the conditional distributions converge to the moments of the ones of the limit, are also derived.