Limit theorems for distributions invariant under groups of transformations
变换群不变分布的极限定理
A distributional symmetry is invariance of a distribution under a group of transformations. Exchangeability and stationarity are examples. We explain that a result of ergodic theory implies a law of large numbers for such invariant distributions: If the group satisfies suitable conditions, expectations can be estimated by averaging over subsets of transformations, and these estimators are strongly consistent. We show that, if a mixing condition holds, the averages also…