A variant of the Skorokhod theorem for Markov sequences (Q2726260)

The following problem is solved with the help of a special representation of the Markov chain as a random evolution. Let \((x_{n}\); \(n\geq 1)\) be a homogeneous Markov chain in the complete separable metric space \((X, r)\) with the transition probability \(Q.\) Consider now the stationary Markov chain \((y_{n}\); \(n\geq 1)\) in \((X, r)\) with the same transition probability \(Q.\) The latter chain can be defined on a different probability space than the former one. The distributions of \(x_{n}\) and \(y_{n}\) converge to each other in variation. Then there exist a probability space and two Markov chains \((x_{n}'; n\geq 1)\) and \((y_{n}'\); \(n\geq 1)\) on it which have the same distributions as the previous two sequences and such that \(\rho(x_{n}', y_{n}')\to 0\) as \(n\to\infty\pmod P\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].