期刊名称：Sankhya. Series A, mathematical statistics and probability
印刷版ISSN：0976-836X
电子版ISSN：0976-8378
出版年度：2010
卷号：72
期号：01
页码：208--220
出版社：Indian Statistical Institute
摘要：Let (¦Ìn : n ≥ 0) be Borel probabilities on a metric space S such that ¦Ìn → ¦Ì0
weakly. Say that Skorohod representation holds if, on some probability space,
there are S-valued random variables Xn satisfying Xn ～ ¦Ìn for all n and
Xn → X0 in probability. By Skorohod¡¯s theorem, Skorohod representation
holds (with Xn → X0 almost uniformly) if ¦Ì0 is separable. Two results
are proved in this paper. First, Skorohod representation may fail if ¦Ì0 is
not separable (provided, of course, non separable probabilities exist). Second,
independently of ¦Ì0 separable or not, Skorohod representation holds if
W(¦Ìn, ¦Ì0) → 0 where W is Wasserstein distance (suitably adapted). The
converse is essentially true as well. Such a W is a version of Wasserstein
distance which can be defined for any metric space S satisfying a mild condition.
To prove the quoted results (and to define W), disintegrable probability
measures are fundamental.
关键词：Disintegration, separable probability measure, Skorohod
representation theorem, Wasserstein distance, weak convergence of
probability measures
