Embedded Markov chain approximations in Skorokhod topologies

TL;DR

This paper explores how different embeddings of Markov chains into continuous processes relate to each other within various Skorokhod topologies, establishing convergence implications and tightness conditions.

Contribution

It demonstrates that convergence in the step function embedding implies convergence in other embeddings and provides conditions for the reverse, linking various Skorokhod topologies.

Findings

Convergence in the $J_1$ step function embedding implies convergence in other embeddings.

A $J_1$-tightness condition is established for the reverse implication.

$J_1$ convergence is equivalent to joint convergence in $M_1$ and $J_2$.

Abstract

In order to approximate a continuous time stochastic process by discrete time Markov chains one has several options to embed the Markov chains into continuous time processes. On the one hand there is the Markov embedding, which uses exponential waiting times. On the other hand each Skorokhod topology naturally suggests a certain embedding. These are the step function embedding for $J_1$, the linear interpolation embedding for $M_1$, the multi step embedding for $J_2$ and a more general embedding for $M_2$. We show that the convergence of the step function embedding in $J_1$ implies the convergence of the other embeddings in the corresponding topologies, respectively. For the converse statement a $J_1$-tightness condition for embedded Markov chains is given. The result relies on various representations of the Skorokhod topologies. Additionally it is shown that $J_1$ convergence is equivalent to the joint convergence in $M_1$ and $J_2$.