Can someone please help me with the following problem?
Let be independent and irreducible time-homogenous Markov Chains with common state-space E, and transition matrices P and Q respectively.
(a) Show that the two-dimentsional process defined by is a time-homogenous Markov chain with state space E E.
(b) Prove that if and are both periodic, then can be reducible.
(c) Show that is irreducible if both and are aperiodic. Hint: Use the fact that if is irreducible and aperiodic, then for each there exists some such that for all