LIMIT THEOREMS FOR DISTRIBUTIONS INVARIANT UNDER GROUPS OF TRANSFORMATIONS

• 成果类型: Article
• 署名作者: Austern, Morgane; Orbanz, Peter
• 署名单位: Harvard University; University of London; University College London
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2165
• 发表日期: 2022
• 页码: 1960-1991
• 关键词: gaussian limits; graphs; models

摘要

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 satisfy a central limit theorem, a Berry-Esseen bound, and concentration. These are extended further to apply to triangular arrays, to randomly subsampled averages, and to a generalization of U-statistics. As applications, we obtain a general limit theorem for exchangeable random structures, and new results on stationary random fields, network models, and a class of marked point processes. We also establish asymptotic normality of the empirical entropy for a large class of processes. Some known results are recovered as special cases, and can hence be interpreted as an outcome of symmetry. The proofs adapt Stein's method.