Daniel Hlubinka
Implicit Markov kernels in probability theory

Comment.Math.Univ.Carolinae 43,3 (2002) 547-564.

Abstract:Having Polish spaces $\Bbb X$, $\Bbb Y$ and $\Bbb Z$ we shall discuss the existence of an $\Bbb X \times \Bbb Y$-valued random vector $(\xi ,\eta )$ such that its conditional distributions $K_{x} = \Cal L(\eta \mid \xi =x)$ satisfy $e(x, K_{x}) = c(x)$ or $e(x,K_{x}) \in C(x)$ for some maps $e:\Bbb X\times \Cal M_1(\Bbb Y) \to \Bbb Z$, $c:\Bbb X \to \Bbb Z$ or multifunction $C:\Bbb X \to 2^{\Bbb Z}$ respectively. The problem is equivalent to the existence of universally measurable Markov kernel $K:\Bbb X \to \Cal M_1(\Bbb Y)$ defined implicitly by $e(x, K_{x}) = c(x)$ or $e(x,K_{x}) \in C(x)$ 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 the theory on the generalized moment problem.

Keywords: Markov kernels, universal measurability, selections, moment problems, extreme points
AMS Subject Classification: 28A35, 28B20, 46A55, 60A10, 60B05