Statistical implications of group invariance of distributions

Abstract: Consider a large random structure – a random graph, a stochastic process on the line, a random field on the grid – and a function that depends only on a small part of the structure. Now use a family of transformations to ‘move’ the domain of the function over the structure, collect each function value, and average. Under suitable conditions, the law of large numbers generalizes to such averages; that is one of the deep insights of modern ergodic theory. My own recent work with Morgane Austern (Harvard) shows that central limit theorems and other higher-order properties also hold. Loosely speaking, if the i.i.d. assumption of classical statistics is substituted by suitable properties formulated in terms of groups, the fundamental theorems of inference still hold.