Mixing and the Central Limit Theorem

Abstract

The object of this chapter is to show how conditions like uniform ergodicity, uniform mixing and related notions allow one to extend certain limit theorems valid for sequences of independent random variables to stationary Markov sequences. In the first section, the case of independence is discussed and the remarkable result of Kolmogorov on the approximation of an nth convolution of a distribution by an infinitely divisible law with error term is obtained. Uniform ergodicity and strong mixing are introduced in section 2. The relation of these to the ordinary concepts of ergodicity and mixing are discussed. Cogburn’s interesting result on limit laws for stationary Markov sequences that are uniformly ergodic is derived. An operator formulation of strong mixing and uniform ergodicity is given in the following section. Various L P norm conditions on the Markov transition operator are also introduced. The L P norm condition is shown to be equivalent to the maximal absolute correlation between past and future tending to zero as the distance between past and future tends to zero. The various conditions are examined in the case of random walks on compact groups. A central limit theorem is proved for stationary Markov sequences in the last section and applied to random walks on compact groups.