Limit theorems for distributions invariant under groups of transformations
The Annals of Statistics · 2022 · Volume 50 · Issue 4
Abstract
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.
Keywords
37A30, 60F05, 60G09, 62G20, 62M99, asymptotic normality, Berry–Esseen theorems, central limit theorems, ergodicity, exchangeability, pointwise ergodic theorems, Stein’s method, symmetry
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