Suppose $\{X\_{n}\}$ and $\{Y\_{n}\}$ are integer-valued Markov chains indexed on nonnegative integers $n$. Is $(X\_{n},Y\_{n})$ a Markov chain? If not, is it a Markov chain when $\{X\_{n}\}$ and $\{Y\_{n}\}$ are independent collections of random variables?