TRIVIAL INTERSECTION OF σ-FIELDS AND GIBBS SAMPLING

• 成果类型: Article
• 署名作者: Berti, Patrizia; Pratelli, Luca; Rigo, Pietro
• 署名单位: Universita di Modena e Reggio Emilia; University of Pavia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/07-AOP387
• 发表日期: 2008
• 页码: 2215-2234
• 关键词: DISTRIBUTIONS

摘要

Let (Omega, F, P) be a probability space and N the class of those F is an element of F satisfying P(F) is an element of {0, 1}. For each G subset of F, define (G) over bar = sigma(G boolean OR N). Necessary and sufficient conditions for (A) over bar boolean AND (B) over bar = (A boolean AND B) over bar, where A, B subset of F 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, F) = (X x Y, U circle times V) where (X, U) and (Y, V) are measurable spaces. Let (X(n), Y(n))(n >= 0) be the Gibbs chain for P. Then, the SLLN holds for (X(n), Y(n)) if and only if <(sigma(X))over bar> boolean AND <(sigma(Y))over bar> = N, or equivalently if and only if P(X is an element of U) P (Y is an element of V) = 0 whenever U is an element of U, V is an element of V and P(U x V) = P(U(c) x V(c)) = 0. The latter condition is also equivalent to ergodicity of (X(n), Y(n)), on a certain subset S(0) subset of Omega, in case F = U circle times V is countably generated and P absolutely continuous with respect to a product measure.