How to understand this kind of Markov chain? There are two unidirectionally coupled processes $X_t$ and $Y_t$. The coupling is $Y_t=g(X_{t-\delta},\dots)$. The Bayesian network of the two process is described in the following figure:

Now this article points out that since $X_{t-\delta-\xi}$ is d-separated from $Y_t$ given the joint variable $(X_{t-\delta},Y_{t-1})$, a Markov chain, $X_{t-\delta-\xi}\rightarrow(X_{t-\delta},Y_{t-1})\rightarrow Y_t$, is formed.
Let us suppose $X_{t-\delta-\xi}\rightarrow(X_{t-\delta},Y_{t-1})\rightarrow Y_t$ is indeed a Markov chain. I have never seen this kind of Markov chain before. The "textbook" version of a Markov chain, $X_{t-1}\rightarrow X_t\rightarrow X_{t+1}$, the past, the current and the future states share the same state space. However, in $X_{t-\delta-\xi}\rightarrow(X_{t-\delta},Y_{t-1})\rightarrow Y_t$, the past, the current and the future states all have different state space.
My question is, how do you simulate (Monte Carlo simulation) this kind of Markov chain?
I mean, for "textbook" Markov chain $X_{t-1}\rightarrow X_t\rightarrow X_{t+1}$, a simulation may generate a realization such as $(1, 2, 2, 4, 3, 2, \dots)$. Can you do that for $X_{t-\delta-\xi}\rightarrow(X_{t-\delta},Y_{t-1})\rightarrow Y_t$?

I think this may be a misunderstanding due to a common abuse of terminology. Typically, a Markov chain is a special type of stochastic process, namely one satisfying the Markov property: If $X$ is a Markov chain, then $X_{t-1}$ and $X_{t+1}$ are conditionally independent given $X_t$.
A Markov tuple is a tuple of random variables that are conditionally independent: For the Markov tuple $X-Y-Z-W$, for example, we have that $X$ and $Z$ are conditionally independent given $Y$, $X$ and $W$ are conditionally independent given either $Y$ or $Z$, and $Y$ and $W$ are conditionally independent given $Z$. In some articles and books, however, a Markov tuple is also called a Markov chain, although it is not a stochastic process but just a collection of random variables with a given conditional independence structure.
Hence, I believe in the article you mention $X_{t-\delta-\xi}-(X_{t-\delta},Y_{t-1})-Y_t$ is a Markov tuple and not a Markov chain in the sense of a stochastic process.
