Implicit Markov kernels in probability theory

Hlubinka, Daniel. "Implicit Markov kernels in probability theory." Commentationes Mathematicae Universitatis Carolinae 43.3 (2002): 547-564. <http://eudml.org/doc/249003>.

@article{Hlubinka2002,
abstract = {Having Polish spaces $\mathbb \{X\}$, $\mathbb \{Y\}$ and $\mathbb \{Z\}$ we shall discuss the existence of an $\mathbb \{X\} \times \mathbb \{Y\}$-valued random vector $(\xi ,\eta )$ such that its conditional distributions $\operatorname\{K\}_\{x\} = \mathcal \{L\}(\eta \mid \xi =x)$ satisfy $e(x, \operatorname\{K\}_\{x\}) = c(x)$ or $e(x,\operatorname\{K\}_\{x\}) \in C(x)$ for some maps $e:\mathbb \{X\}\times \mathcal \{M\}_1(\mathbb \{Y\}) \rightarrow \mathbb \{Z\}$, $c:\mathbb \{X\} \rightarrow \mathbb \{Z\}$ or multifunction $C:\mathbb \{X\} \rightarrow 2^\{\mathbb \{Z\}\}$ respectively. The problem is equivalent to the existence of universally measurable Markov kernel $\operatorname\{K\}:\mathbb \{X\} \rightarrow \mathcal \{M\}_1(\mathbb \{Y\})$ 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.},
author = {Hlubinka, Daniel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Markov kernels; universal measurability; selections; moment problems; extreme points; Markov kernels; selections; moment problems; extreme points},
language = {eng},
number = {3},
pages = {547-564},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Implicit Markov kernels in probability theory},
url = {http://eudml.org/doc/249003},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Hlubinka, Daniel
TI - Implicit Markov kernels in probability theory
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 3
SP - 547
EP - 564
AB - Having Polish spaces $\mathbb {X}$, $\mathbb {Y}$ and $\mathbb {Z}$ we shall discuss the existence of an $\mathbb {X} \times \mathbb {Y}$-valued random vector $(\xi ,\eta )$ such that its conditional distributions $\operatorname{K}_{x} = \mathcal {L}(\eta \mid \xi =x)$ satisfy $e(x, \operatorname{K}_{x}) = c(x)$ or $e(x,\operatorname{K}_{x}) \in C(x)$ for some maps $e:\mathbb {X}\times \mathcal {M}_1(\mathbb {Y}) \rightarrow \mathbb {Z}$, $c:\mathbb {X} \rightarrow \mathbb {Z}$ or multifunction $C:\mathbb {X} \rightarrow 2^{\mathbb {Z}}$ respectively. The problem is equivalent to the existence of universally measurable Markov kernel $\operatorname{K}:\mathbb {X} \rightarrow \mathcal {M}_1(\mathbb {Y})$ 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.
LA - eng
KW - Markov kernels; universal measurability; selections; moment problems; extreme points; Markov kernels; selections; moment problems; extreme points
UR - http://eudml.org/doc/249003
ER -