Splitting of liftings in products of probability spaces
| dc.creator | Strauss, W. | |
| dc.creator | Macheras, N. D. | |
| dc.creator | Musial, K. | |
| dc.date | 2005-09-01 | |
| dc.date.accessioned | 2026-07-07T05:22:52Z | |
| dc.date.available | 2026-07-07T05:22:52Z | |
| dc.description | We prove that if (X,\mathfrakA,P) is an arbitrary probability space with countably generated σ-algebra \mathfrakA, (Y,\mathfrakB,Q) is an arbitrary complete probability space with a lifting ρand \hat R is a complete probability measure on \mathfrakA \hat \otimes_R \mathfrakB determined by a regular conditional probability {S_y:y\in Y} on \mathfrakA with respect to \mathfrakB, then there exist a lifting πon (X\times Y,\mathfrakA \hat \otimes_R \mathfrakB,\hat R) and liftings σ_y on (X,\hat \mathfrakA_y,\hat S_y), y\in Y, such that, for every E\in\mathfrakA \hat \otimes_R \mathfrakB and every y\in Y, [π(E)]^y=σ_y\bigl([π(E)]^y\bigr). Assuming the absolute continuity of R with respect to P\otimes Q, we prove the existence of a regular conditional probability {T_y:y\in Y} and liftings \varpi on (X\times Y,\mathfrakA \hat \otimes_R \mathfrakB,\hat R), ρ' on (Y,\mathfrakB,\hat Q) and σ_y on (X,\hat \mathfrakA_y,\hat S_y), y\in Y, such that, for every E\in\mathfrakA \hat \otimes_R \mathfrakB and every y\in Y, [\varpi(E)]^y=σ_y\bigl([\varpi(E)]^y\bigr) and \varpi(A\times B)=\bigcup_{y\inρ'(B)}σ_y(A)\times{y}\qquadif A\times B\in\mathfrakA\times\mathfrakB. 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 \hat R-measurable stochastic processes into their \hat R-measurable modifications. | |
| dc.description | Published at http://dx.doi.org/10.1214/009117904000000018 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/math/0509010 | |
| dc.identifier | http://arxiv.org/abs/math/0509010 | |
| dc.identifier | Annals of Probability 2004, Vol. 32, No. 3, 2389-2408 | |
| dc.identifier | doi:10.1214/009117904000000018 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/76227 | |
| dc.subject | Probability | |
| dc.subject | 28A51, 28A50, 28A35, 60A01, 60G05. (Primary) | |
| dc.title | Splitting of liftings in products of probability spaces | |
| dc.type | text |