Asymptotic variance of functionals of discrete-time Markov chains via the Drazin inverse.

Dan J. Spitzner (Department of Statistics (0439), Virginia Tech, Blacksburg, VA)
Thomas R Boucher (Department of Mathematics, Plymouth State)

Abstract


We consider a $\psi$-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.

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Pages: 120-133

Publication Date: April 24, 2007

DOI: 10.1214/ECP.v12-1262

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