Asymptotic Variance of Functionals of Discrete- Time Markov Chains via the Drazin Inverse

We consider a ψ-irreducible, discrete-time Markov chain on a general state space with transition kernel P . Under suitable conditions on the chain, kernels can be treated as bounded linear operators between spaces of functions or measures and the Drazin inverse of the kernel operator I−P exists. The Drazin inverse provides a unifying framework for objects governing the chain. This framework is applied to derive a computational technique for the asymptotic variance in the central limit theorems of univariate and higher-order partial sums. Higher-order partial sums are treated as univariate sums on a ‘sliding-window’ chain. Our results are demonstrated on a simple AR(1) model and suggest a potential for computational simplification.