566 research outputs found
Splitting of liftings in products of probability spaces
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
Arteriovenous Malformation of the Pancreas
Pancreatic arteriovenous malformation (PAVM) is a very rare and mostly congenital lesion, with less than 80 cases described in the English-published literature. It is defined as a tumorous vascular abnormality that is constructed between an anomalous bypass anastomosis of the arterial and venous networks within the pancreas. It represents about 5% of all arteriovenous malformations found in the gastrointestinal tract. Herein, we present a 64-year-old patient with symptomatic PAVM involving the body and tail of the organ, which was successfully treated by transcatheter arterial embolization. The disease spectrum and review of the literature are also presented
Splitting of liftings in products of probability spaces II
AbstractFor a probability measure R on a product of two probability spaces that is absolutely continuous with respect to the product measure we prove the existence of liftings subordinated to a regular conditional probability and the existence of a lifting for R with lifted sections which satisfies in addition a rectangle formula. These results improve essentially some of the results from the former work of the authors [W. Strauss, N.D. Macheras, K. Musiał, Splitting of liftings in products of probability spaces, Ann. Probab. 32 (2004) 2389–2408], by weakening considerably the assumptions and by presenting more direct and shorter proofs. In comparison with [W. Strauss, N.D. Macheras, K. Musiał, Splitting of liftings in products of probability spaces, Ann. Probab. 32 (2004) 2389–2408] it is crucial for applications intended that we can now prescribe one of the factor liftings completely freely. We demonstrate the latter by applications to τ-additive measures, transfer of strong liftings, and stochastic processes
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