## Electronic Communications in Probability

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

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 12 (2007), paper no. 13, 120-133.

Dates
Accepted: 24 April 2007
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ecp/1465224956

Digital Object Identifier
doi:10.1214/ECP.v12-1262

Mathematical Reviews number (MathSciNet)
MR2300221

Zentralblatt MATH identifier
1128.60062

Rights

#### Citation

Spitzner, Dan; Boucher, Thomas. Asymptotic variance of functionals of discrete-time Markov chains via the Drazin inverse. Electron. Commun. Probab. 12 (2007), paper no. 13, 120--133. doi:10.1214/ECP.v12-1262. https://projecteuclid.org/euclid.ecp/1465224956

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