On mixing and the central limit theorem

The Central Limit Theorem for uniformly strong mixing measures

The theorem of Shannon-McMillan-Breiman states that for every generating partition on an ergodic system, the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere (provided the entropy is finite). In this paper we prove that the measure of cylinder sets are lognormally distributed for strongly mixing systems and infinite partitions and show that the ...

A Central Limit Theorem and a Strong Mixing Condition.

A Lyapunov type central limit theorem for mixing quantum systems

Physical systems, composed of interacting identical (or similar) subsystems appear in many branches of physics. They are standard in condensed matter physics. Assuming that each subsystem only interacts with its nearest neighbours and that the energy per subsystem has an upper limit, which must not depend on the number of subsystems n, we show that the distribution of energy eigenvalues of almo...

On the Markov chain central limit theorem

The goal of this paper is to describe conditions which guarantee a central limit theorem for functionals of general state space Markov chains. This is done with a view towards Markov chain Monte Carlo settings and hence the focus is on the connections between drift and mixing conditions and their implications. In particular, we consider three commonly cited central limit theorems and discuss th...

Variations on the Projective Central Limit Theorem

This expository article states and proves four, concrete, projective, central limit theorems. The results are known or suspected to be true by experts who are familiar with the more general central limit theorem for convex bodies, and related theory. Here we consider only four types of high dimensional geometric objects: spheres, balls, cubes, and boundaries of cubes. Each is capable of transfo...