On the Markov chain central limit theorem

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...

Convergence of Moments in a Markov-chain Central Limit Theorem

Let (Xi)1i=0 be a V -uniformly ergodic Markov chain on a general state space, and let be its stationary distribution. For g : X! R, de ne Wk(g) := k 1=2 k 1 X i=0 g(Xi) (g) : It is shown that if jgj V 1=n for a positive integer n, then ExWk(g) n converges to the n-th moment of a normal random variable with expectation 0 and variance 2 g := (g ) (g) + 1 X j=1 Z g(x)Exg(Xj) (g) 2 : This extends t...

Variance Estimation in the Central Limit Theorem for Markov Chains

This article concerns the variance estimation in the central limit theorem for finite recurrent Markov chains. The associated variance is calculated in terms of the transition matrix of the Markov chain. We prove the equivalence of different matrix forms representing this variance. The maximum likelihood estimator for this variance is constructed and it is proved that it is strongly consistent ...

A central limit theorem for functions of a Markov chain with applications to shifts

A Functional Central Limit Theorem for a Class of Interacting Markov Chain Monte Carlo Methods

We present a functional central limit theorem for a new class of interacting Markov chain Monte Carlo algorithms. These stochastic algorithms have been recently introduced to solve non-linear measure-valued equations. We provide an original theoretical analysis based on semigroup techniques on distribution spaces and fluctuation theorems for self-interacting random fields. Additionally we also ...