Open Access
November, 1983 Orthogonal Transition Kernels
R. Daniel Mauldin, David Preiss, Heinrich v. Weizsacker
Ann. Probab. 11(4): 970-988 (November, 1983). DOI: 10.1214/aop/1176993446


A transition kernel $(\mu_x)_{x \in X}$ between Polish spaces $X$ and $Y$ is called completely orthogonal if the $\mu_x$ are separated by the fibers of a Borel map $\varphi: Y \rightarrow X$. It is orthogonality preserving if orthogonal measures on $X$ induce orthogonal mixtures on $Y$. We give a von Neumann "type" isomorphism theorem for atomless completely orthogonal kernels, and a theorem and some counterexamples concerning the separation of two orthogonal measure convex sets of probability measures by a measurable set. These techniques yield three results on orthogonality preserving kernels: (1) They need not be completely orthogonal but (2) are uniformly orthogonal (in the sense of D. Maharam) and (3) if $X$ is $\sigma$-compact, $Y = \lim_\leftarrow Y_n$ and $(\mu_x)$ is orthogonality preserving and continuous in $x$ then there is even a strongly consistent sequence of statistics $\varphi_n: Y_n \rightarrow X$ for $(\mu_x)$.


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R. Daniel Mauldin. David Preiss. Heinrich v. Weizsacker. "Orthogonal Transition Kernels." Ann. Probab. 11 (4) 970 - 988, November, 1983.


Published: November, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0528.60006
MathSciNet: MR714960
Digital Object Identifier: 10.1214/aop/1176993446

Primary: 60A10
Secondary: 28A75

Keywords: classification of kernels , filters of countable type , Orthogonal measures , perfect statistics , simultaneous Lebesgue decompositions

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 4 • November, 1983
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