Implicit Markov kernels in probability theory

Hlubinka, Daniel

Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 3, pp. 547-564.

Résumé

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 $\operatorname{K}_{x} = \Cal L(\eta\mid \xi=x)$ satisfy $e(x, \operatorname{K}_{x}) = c(x)$ or $e(x,\operatorname{K}_{x}) \in C(x)$ 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 $\operatorname{K}:\Bbb X \to \Cal M_1(\Bbb Y)$ defined implicitly by $e(x, \operatorname{K}_{x}) = c(x)$ or $e(x,\operatorname{K}_{x}) \in C(x)$ 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.

Détail

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