Trivial intersection of $σ$-fields and Gibbs sampling
| dc.creator | Berti, Patrizia | |
| dc.creator | Pratelli, Luca | |
| dc.creator | Rigo, Pietro | |
| dc.date | 2009-01-19 | |
| dc.date.accessioned | 2026-07-07T12:31:37Z | |
| dc.date.available | 2026-07-07T12:31:37Z | |
| dc.description | Let $(Ω,\mathcal{F},P)$ be a probability space and $\mathcal{N}$ the class of those $F\in\mathcal{F}$ satisfying $P(F)\in\{0,1\}$. For each $\mathcal{G}\subset\mathcal{F}$, define $\overline{\mathcal{G}}=σ(\mathcal{G}\cup\mathcal {N})$. Necessary and sufficient conditions for $\overline{\mathcal{A}}\cap\overline{\mathcal{B}}=\overline {\mathcal {A}\cap\mathcal{B}}$, where $\mathcal{A},\mathcal{B}\subset\mathcal{F}$ are sub-$σ$-fields, are given. These conditions are then applied to the (two-component) Gibbs sampler. Suppose $X$ and $Y$ are the coordinate projections on $(Ω,\mathcal{F})=(\mathcal{X}\times\mathcal{Y},\mathcal {U}\otimes \mathcal{V})$ where $(\mathcal{X},\mathcal{U})$ and $(\mathcal{Y},\mathcal{V})$ are measurable spaces. Let $(X_n,Y_n)_{n\geq0}$ be the Gibbs chain for $P$. Then, the SLLN holds for $(X_n,Y_n)$ if and only if $\overline{σ(X)}\cap\overline{σ(Y)}=\mathcal{N}$, or equivalently if and only if $P(X\in U)P(Y\in V)=0$ whenever $U\in\mathcal{U}$, $V\in\mathcal{V}$ and $P(U\times V)=P(U^c\times V^c)=0$. The latter condition is also equivalent to ergodicity of $(X_n,Y_n)$, on a certain subset $S_0\subsetΩ$, in case $\mathcal{F}=\mathcal{U}\otimes\mathcal{V}$ is countably generated and $P$ absolutely continuous with respect to a product measure. | |
| dc.description | Published in at http://dx.doi.org/10.1214/07-AOP387 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/0901.2851 | |
| dc.identifier | http://arxiv.org/abs/0901.2851 | |
| dc.identifier | Annals of Probability 2008, Vol. 36, No. 6, 2215-2234 | |
| dc.identifier | doi:10.1214/07-AOP387 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/216504 | |
| dc.subject | Probability | |
| dc.subject | 60A05, 60A10, 60J22, 65C05 (Primary) | |
| dc.title | Trivial intersection of $σ$-fields and Gibbs sampling | |
| dc.type | text |