THE CENTRAL LIMIT THEOREM FOR UNIFORMLY STRONG MIXING MEASURES

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...

A regeneration proof of the central limit theorem for uniformly ergodic Markov chains

E h(x)π(dx). Ibragimov and Linnik (1971) proved that if (Xn) is geometrically ergodic, then a central limit theorem (CLT) holds for h whenever π(|h|) < ∞, δ > 0. Cogburn (1972) proved that if a Markov chain is uniformly ergodic, with π(h) < ∞ then a CLT holds for h. The first result was re-proved in Roberts and Rosenthal (2004) using a regeneration approach; thus removing many of the technicali...

Strong Law of Large Numbers and Functional Central Limit Theorem

20.1. Additional technical results on weak convergence Given two metric spaces S1, S2 and a measurable function f : S1 → S2, suppose S1 is equipped with some probability measure P. This induces a probability measure on S2 which is denoted by Pf−1 and is defined by Pf−1(A) = P(f−1(A) for every measurable set A ⊂ S2. Then for any random variable X : S2 → R, its expectation EPf −1 [X] is equal to ...