Suppose $X(t)$ is a Markov chain taking values in $\{0,1\}^2$. Suppose $q$ is the q matrix whose positive valued entries are
$q((0,0),(1,0)) = \beta_{0}$
$q((1,0),(0,0)) = \delta_{0}$
$q((0,1),(1,1)) = \beta_{0}$
$q((1,1),(0,1)) = \delta_{0}$
$q((0,0),(0,1)) = \beta_{1}$
$q((0,1),(0,0)) = \delta_{1}$
$q((1,0),(1,1)) = \beta_{1}$
$q((1,1),(1,0)) = \delta_{1}$
How do I rigorously show that the random processes $X_{1}(t)$ and $X_{2}(t)$ are independent random processes? Here $X_{i}(t)$ means the $i-th$ component of $X(t)$
First compute carefully the marginals.
Then assuming the initial states are independant, you make sure that the product of marginals has the same law as the whole chain, which is straigtforward.