Limit theorems for invariant distributions

Abstract

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.

A free central-limit theorem for dynamical systems

“…We note that the Berry-Esseen bound depends on how fast the free mixing coefficients (ℵ n [b|G n ]) decrease as a function of b. Notably if ℵ n [b|G n ] = 0 for all b > 0 then we obtain 4 .…”

“…In this section we present a known result ( see e.g [4]), that will be used in the proof of proposition 4, proposition 10 and proposition 11.…”

“…Central-limit theorems have been extended to dependent sequences by enforcing conditions on the strong mixing coefficients [8]. This notion has been generalized to general dynamical systems (Z g ) defined by a group action on a random object [4]. This is done by choosing a metric on the underlying group and upper-bounding the correlations between events depending on {Z g 1 , Z g 2 } and events depending on {Z g , g ∈ G} for a subset G "far away " from g 1 and g 2 .…”

“…Moreover the free central limit theorem exclusively studies empirical averages and not more complex quantities such as U-statistics of which i,j≤n Φ(X i , X j ) (for a measurable function Φ) is an example. This represents a disconnect with what is known for the classical central-limit theorem where the assumption of independence has been successfully relaxed for general dynamical systems [4,8], and where the limit of U-statistics are extensively studied. This paper bridges this gap.…”

Mixing Properties of Stable Random Fields Indexed by Amenable and Hyperbolic Groups

“…Random fields indexed by amenable groups (e.g., Heisenberg groups, discrete matrix groups, group of permutations of N with finite support, etc.) arise naturally in machine learning algorithms for structured and dependent space-time data [3]. On the other hand, random fields indexed by hyperbolic groups are useful in tree-indexed processes, branching models, etc.…”