Splitting of liftings in products of probability spaces

Splitting of liftings in products of probability spaces

N. D. Macheras, K. Musiał, and W. Strauss

Source: Ann. Probab. Volume 32, Number 3B (2004), 2389-2408.

Abstract

We prove that if $(X,{\mathfrak{A}},P)$ is an arbitrary probability space with countably generated σ-algebra ${\mathfrak{A}}$ , $(Y,{\mathfrak{B}},Q)$ is an arbitrary complete probability space with a lifting ρ and $\widehat {R}$ is a complete probability measure on ${\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ determined by a regular conditional probability {S_y:y∈Y} on ${\mathfrak{A}}$ with respect to ${\mathfrak{B}}$ , then there exist a lifting π on $(X\times Y,{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}},\widehat {R})$ and liftings σ_y on $(X,\widehat {\mathfrak{A}}_{y},\widehat {S}_{y})$ , y∈Y, such that, for every $E\in{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ and every y∈Y, $[\pi(E)]^{y}=\sigma_{y}\bigl([\pi(E)]^{y}\bigr).$ Assuming the absolute continuity of R with respect to P⊗Q, we prove the existence of a regular conditional probability {T_y:y∈Y} and liftings ϖ on $(X\times Y,{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}},\widehat {R})$ , ρ' on $(Y,\mathfrak{B},\widehat {Q})$ and σ_y on $(X,\widehat {\mathfrak{A}}_{y},\widehat {S}_{y})$ , y∈Y, such that, for every $E\in{\mathfrak{A}}{\,\widehat{\otimes}_{R}\,}{\mathfrak{B}}$ and every y∈Y, $[\varpi(E)]^{y}=\sigma_{y}\bigl([\varpi(E)]^{y}\bigr)$ and $\varpi(A\times B)=\bigcup_{y\in\rho'(B)}\sigma_{y}(A)\times\{y\}\qquad\mbox{if }A\times B\in{\mathfrak{A}}\times{\mathfrak{B}}.$ Both results are generalizations of Musiał, 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 $\widehat {R}$ -measurable stochastic processes into their $\widehat {R}$ -measurable modifications.

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Primary Subjects: 28A51, 28A50, 28A35, 60A01, 60G05

Keywords: Liftings; product liftings; product measures; regular conditional probabilities; densities; product densities; measurable stochastic processes

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1091813617
Digital Object Identifier: doi:10.1214/009117904000000018
Mathematical Reviews number (MathSciNet): MR2078544
Zentralblatt MATH identifier: 1058.60005

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