Product liftings and densities with lifting invariant and density invariant sections

Musiał, Kazimierz; Strauss, W.; Macheras, N. D.

doi:10.4064/fm-166-3-281-303

Cited by 10 publications

(5 citation statements)

References 12 publications

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“…The next result corresponds to Lemma 2.1 of [10] and plays an essential role in the proof of Proposition 3.4.…”

Section: Arbitrary Probability Measuresmentioning

confidence: 73%

“…Such a property holds true for the density ψ in Theorem 3.5. As it has been observed in [10], an improvement of this type is in general impossible in case of product measures R.…”

Section: Remark 37mentioning

confidence: 86%

“…We show that in this case one can achieve (SP) (see Theorem 3.6). The proof is completely independent of [10].…”

mentioning

confidence: 86%

“…The whole paper consists of two independent parts. The first one is a continuation of [10]. We have proven in [10] for complete probability spaces (X, A, P ) and (Y, B, Q) that, for given lifting ρ ∈ Λ(Q), there exist liftings σ ∈ Λ(P ) and π ∈ Λ(P ⊗ Q) such that…”

mentioning

confidence: 99%

“…Throughout this section, probability spaces (X, A, P ) and (Y, B, Q) are arbitrary but (Y, B, Q) is assumed to be complete. According to [10], Theorem 2.9, for given density ρ ∈ ϑ(Q), there exist densities σ ∈ ϑ(P ) and ϕ ∈ ϑ(P ⊗ Q) such that, for all E ∈ A ⊗ B, we have:…”

mentioning

confidence: 99%

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Splitting of liftings in products of probability spaces

Strauss¹,

Macheras²,

Musiał³

2004

Ann. Probab.

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We prove that if (X,\mathfrakA,P) is an arbitrary probability space with countably generated \sigma-algebra \mathfrakA, (Y,\mathfrakB,Q) is an arbitrary complete probability space with a lifting \rho and \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 \pi on (X\times Y,\mathfrakA \hat \otimes_R \mathfrakB,\hat R) and liftings \sigma_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, [\pi(E)]^y=\sigma_y\bigl([\pi(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), \rho' on (Y,\mathfrakB,\hat Q) and \sigma_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=\sigma_y\bigl([\varpi(E)]^y\bigr) and \varpi(A\times B)=\bigcup_{y\in\rho'(B)}\sigma_y(A)\times{y}\qquadif A\times B\in\mathfrakA\times\mathfrakB. Both results are generalizations of Musia\l, 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.Comment: 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

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“…The next result corresponds to Lemma 2.1 of [10] and plays an essential role in the proof of Proposition 3.4.…”

Section: Arbitrary Probability Measuresmentioning

confidence: 73%

“…Such a property holds true for the density ψ in Theorem 3.5. As it has been observed in [10], an improvement of this type is in general impossible in case of product measures R.…”

Section: Remark 37mentioning

confidence: 86%

“…We show that in this case one can achieve (SP) (see Theorem 3.6). The proof is completely independent of [10].…”

mentioning

confidence: 86%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Splitting of liftings in products of probability spaces

Strauss¹,

Macheras²,

Musiał³

2004

Ann. Probab.

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Existence of linear liftings with invariant sections in product measure spaces

2004

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D e d i c a t e d t o P r o f. D r. D. Kö l z o w o n t h e o c c a s i o n o f h i s 75 th b i r t h d a yAbstract. We prove that on a complete product of two arbitrary probability spaces, one of which is endowed with a particular linear lifting (we call it admissible), there exists a linear lifting possessing the property that all its sections determined by the marginal space without lifting are invariant under corresponding, given a priori, marginal linear lifting. If both marginal spaces are endowed with linear liftings (at least one of them to be admissible) then the lifting in the product space may have additional product properties. We prove also that for non-atomic marginal spaces there exist no marginal linear liftings and no linear lifting in the product possessing all its sections invariant under both marginal linear liftings.

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