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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...

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