Let \$(Omega,mathcalF,P)\$ be a probability space and \$mathcalN\$ the class of those \$FinmathcalF\$ satisfying \$P(F)in,1\$. For each \$mathcalGsubsetmathcalF\$, define \$overlinemathcalG=sigmaigl(mathcalGcupmathcalNigr)\$. Necessary and sufficient conditions for \$overlinemathcalAcapoverlinemathcalB=overlinemathcalAcapmathcalB\$, where \$mathcalA,mathcalBsubsetmathcalF\$ are sub-\$sigma\$-fields, are given. These conditions are then applied to the (two component) Gibbs sampler. Suppose \$X\$ and \$Y\$ are the coordinate projections on \$(Omega,mathcalF)=(mathcalX imesmathcalY,mathcalUotimesmathcalV)\$ where \$(mathcalX,mathcalU)\$ and \$(mathcalY,mathcalV)\$ are measurable spaces. Let \$(X_n,Y_n)_ngeq 0\$ be the Gibbs-chain for \$P\$. Then, the SLLN holds for \$(X_n,Y_n)\$ if and only if \$overlinesigma(X)capoverlinesigma(Y)=mathcalN\$, or equivalently if and only if \$P(Xin U)P(Yin V)=0\$ whenever \$UinmathcalU\$, \$VinmathcalV\$ and \$P(U imes V)=P(U^c imes V^c)=0\$. The latter condition is also equivalent to ergodicity of \$(X_n,Y_n)\$, on a certain subset \$S_0subsetOmega\$, in case \$mathcalF=mathcalUotimesmathcalV\$ is countably generated and \$P\$ absolutely continuous with respect to a product measure.

### Trivial intersection of sigma-fields and Gibbs sampling

#### Abstract

Let \$(Omega,mathcalF,P)\$ be a probability space and \$mathcalN\$ the class of those \$FinmathcalF\$ satisfying \$P(F)in,1\$. For each \$mathcalGsubsetmathcalF\$, define \$overlinemathcalG=sigmaigl(mathcalGcupmathcalNigr)\$. Necessary and sufficient conditions for \$overlinemathcalAcapoverlinemathcalB=overlinemathcalAcapmathcalB\$, where \$mathcalA,mathcalBsubsetmathcalF\$ are sub-\$sigma\$-fields, are given. These conditions are then applied to the (two component) Gibbs sampler. Suppose \$X\$ and \$Y\$ are the coordinate projections on \$(Omega,mathcalF)=(mathcalX imesmathcalY,mathcalUotimesmathcalV)\$ where \$(mathcalX,mathcalU)\$ and \$(mathcalY,mathcalV)\$ are measurable spaces. Let \$(X_n,Y_n)_ngeq 0\$ be the Gibbs-chain for \$P\$. Then, the SLLN holds for \$(X_n,Y_n)\$ if and only if \$overlinesigma(X)capoverlinesigma(Y)=mathcalN\$, or equivalently if and only if \$P(Xin U)P(Yin V)=0\$ whenever \$UinmathcalU\$, \$VinmathcalV\$ and \$P(U imes V)=P(U^c imes V^c)=0\$. The latter condition is also equivalent to ergodicity of \$(X_n,Y_n)\$, on a certain subset \$S_0subsetOmega\$, in case \$mathcalF=mathcalUotimesmathcalV\$ is countably generated and \$P\$ absolutely continuous with respect to a product measure.
##### Scheda breve Scheda completa Scheda completa (DC)
2008
Berti P.; Pratelli L.; Rigo P.
File in questo prodotto:
Eventuali allegati, non sono esposti

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/734961`
##### Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

• ND
• 5
• 4