Markov chains and de-initializing processes (Q2771553)

A random sequence \(Y_n\) is called a de-initializing process for a Markov chain \(X_n\) if \(\Pr(X_n\in A \mid X_0,Y_n)=\Pr(X_n\in A\mid Y_n)\) i.e. the conditional distribution of \(X_n\) given \(X_0\) and \(Y_n\) does not depend on \(X_0\). Let \(L(X_n \mid X_0\sim\mu)\) be the distribution of \(X_n\) with the initial \((X_0)\) distribution \(\mu\), and let \(\|\cdot\|\) be the total variation distance. The main result is that if \(Y_n\) is de-initializing for \(X_n\), then NEWLINE\[NEWLINE \|L(X_n|X_0\sim\mu)-L(X_n \mid X_0\sim\mu')\|\leq \|L(Y_n \mid X_0\sim\mu)-L(Y_n \mid X_0\sim\mu')\|. NEWLINE\]NEWLINE The authors use this result to obtain ergodic theorems for different Markov chains, slice and Gibbs samplers. Some concepts similar to the ``de-initializing'' property are considered and relations between them are described.