Limit theorems for invariant distributions
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We consider random processes whose distribution satisfies a symmetry property. Examples of such properties include exchangeability, stationarity, and various others. We show that, under a suitable mixing condition, estimates computed as ergodic averages of such processes satisfy a central limit theorem, a Berry-Esseen bound, and a concentration inequality. These are generalized further to triangular arrays, to a class of generalized U-statistics, and to a form of random censoring. As applications, we obtain new results on exchangeability, and on estimation in random fields and certain network models; extend results on graphon models to stochastic block models with a growing number of classes; give a simpler proof of a recent central limit theorem for marked point processes; and establish asymptotic normality of the empirical entropy of a large class of processes. In certain special cases, we recover well-known properties, which can hence be interpreted as a direct consequence of symmetry. The proofs adapt Stein's method.
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