Splitting of liftings in products of probability spaces

• 成果类型: Article
• 署名作者: Strauss, W; Macheras, ND; Musial, K
• 署名单位: University of Stuttgart; University of Wroclaw
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2004
• 页码: 2389-2408
• 关键词: stochastic-processes

摘要

We prove that if (X, U, P) is an arbitrary probability space with countably generated sigma-algebra U, (Y, B, Q) is an arbitrary complete probability space with a lifting rho and (R) over cap is a complete probability measure on Ucircle times(R) B determined by a regular conditional probability {S-y : y is an element of Y} on U with respect to B, then there exist a lifting pi on (X x Y, Ucircle times(R) B, (R) over cap) and liftings sigma(y) on (X, (U) over cap (y), (S) over cap (y)), y is an element of Y, such that, for every E is an element of U circle times(R) B and every y is an element of Y. [pi(E)](y) = sigma(y)([pi(E)](y)). Assuming the absolute continuity of R with respect to P circle times Q, we prove the existence of a regular conditional probability {T-y : y is an element ofY} and liffings pi on (X x Y, U circle times(R) B, (R) over cap), rho' on (Y, B, (Q) over cap) and sigma(y) on (X, (U) over cap (y), (S) over cap (y)), y is an element of Y, such that, for every E E 121)R T and every y is an element of Y, [pi (E)](y) = sigma(y)([pi(E)](y)) and pi (A x B) = boolean ORyis an element ofrho'(B) sigma(y)(A) x {y} if A x B is an element of U x B. Both results are generalizations of Musial, Strauss and Macheras [Fund. Math. 166 (2000) 281-303] to the case of measures which are not necessarily products of marginal measures. We prove also that liftings obtained in this paper always convert (R) over cap -measurable stochastic processes into their (R) over cap -measurable modifications.