Coupling of Markov chains by randomized stopping times. I: Couplings, harmonic functions and the Poisson equation
We consider a Markov chain on (E,\({\mathcal B})\) generated by a Markov kernel P. We study the question, when we can find for two initial distributions \(\nu\) and \(\mu\) two randomized stopping times T of \((_{\nu}X_ n)_{n\in N}\) and S of \((_{\mu}X_ n)_{n\in N}\), such that the distribution of \(_{\nu}X_ T\) equals the one of \(_{\mu}X_ S\) and T, S are both finite. The answer is given in terms of \(<\nu -\mu,h>\) with h bounded harmonic, or in terms of \(\lim_{n\to \infty}\| 1/(n+1)\sum^{n}_{0}(\nu - \mu)P^ k\|\). For stopping times S, T for two chains \((_{\nu}X_ n)_{n\in N}\), \((_{\mu}X_ n)_{n\in N}\) we consider measures \(\eta\), \(\xi\) on (E,\({\mathcal B})\) defined as follows: \(\eta (A)= \exp ected\) number of visits of \((_{\nu}X_ n)\) to A before T, \(\xi (A)= \exp ected\) number of visits of \((_{\mu}X_ n)\) to A before S. We show that we can construct S, T such that \(\eta\) and \(\xi\) are mutually singular and \({\mathcal L}(_{\nu}X_ T)={\mathcal L}(_{\mu}X_ S)\). We relate \(\eta\) and \(\xi\) to the positive and negative part of certain solutions of the Poisson equation \((I-P)(\cdot)=\nu -\mu.\)
scientific article; zbMATH DE number 3960702
scientific article