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Implicit Markov kernels in probability theory

Daniel Hlubinka (2002)

Commentationes Mathematicae Universitatis Carolinae

Having Polish spaces 𝕏 , 𝕐 and we shall discuss the existence of an 𝕏 × 𝕐 -valued random vector ( ξ , η ) such that its conditional distributions K x = ( η ξ = x ) satisfy e ( x , K x ) = c ( x ) or e ( x , K x ) C ( x ) for some maps e : 𝕏 × 1 ( 𝕐 ) , c : 𝕏 or multifunction C : 𝕏 2 respectively. The problem is equivalent to the existence of universally measurable Markov kernel K : 𝕏 1 ( 𝕐 ) defined implicitly by e ( x , K x ) = c ( x ) or e ( x , K x ) C ( x ) respectively. In the paper we shall provide sufficient conditions for the existence of the desired Markov kernel. We shall discuss some special solutions of the ( e , c ) - or ( e , C ) -problem and illustrate...

Infinite ergodic index d -actions in infinite measure

E. Muehlegger, A. Raich, C. Silva, M. Touloumtzis, B. Narasimhan, W. Zhao (1999)

Colloquium Mathematicae

We construct infinite measure preserving and nonsingular rank one d -actions. The first example is ergodic infinite measure preserving but with nonergodic, infinite conservative index, basis transformations; in this case we exhibit sets of increasing finite and infinite measure which are properly exhaustive and weakly wandering. The next examples are staircase rank one infinite measure preserving d -actions; for these we show that the individual basis transformations have conservative ergodic Cartesian...

Infinite Iterated Function Systems: A Multivalued Approach

K. Leśniak (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

We prove that a compact family of bounded condensing multifunctions has bounded condensing set-theoretic union. Compactness is understood in the sense of the Chebyshev uniform semimetric induced by the Hausdorff distance and condensity is taken w.r.t. the Hausdorff measure of noncompactness. As a tool, we present an estimate for the measure of an infinite union. Then we apply our result to infinite iterated function systems.

Infinitely divisible cylindrical measures on Banach spaces

Markus Riedle (2011)

Studia Mathematica

In this work infinitely divisible cylindrical probability measures on arbitrary Banach spaces are introduced. The class of infinitely divisible cylindrical probability measures is described in terms of their characteristics, a characterisation which is not known in general for infinitely divisible Radon measures on Banach spaces. Further properties of infinitely divisible cylindrical measures such as continuity are derived. Moreover, the classification result enables us to deduce new results on...

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