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 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.
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