EMBEDDED MARKOV CHAIN APPROXIMATIONS IN SKOROKHOD
Abstract: We prove a -tightness condition for embedded Markov chains and discuss four
Skorokhod topologies in a unified manner.
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 a 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 , the linear interpolation embedding for
, the multistep embedding for and a more general embedding for .
We show that the convergence of the step function embedding in implies the
convergence of the other embeddings in the corresponding topologies. For the converse
statement, a -tightness condition for embedded time-homogeneous Markov chains is
Additionally, it is shown that convergence is equivalent to the joint convergence in
2000 AMS Mathematics Subject Classification: Primary: 60B10; Secondary: 60J75,
Keywords and phrases: Markov chain embedding, tightness, Skorokhod space,
Skorokhod topologies, jump processes, Markov chain approximation.