A die is rolled repeatedly. Determine whether the sequence:
"at time n the time Xn of the most recent appearance of a six"
is a Markov Chain.
/// My attempt: in class it was explained that a Markov Chain is homogeneous iff it depends only on x and y and is the same for all n≥1: Pr(X_n =x│X_(n-1) =y).
But, this sequence does not seem homogeneous and since I have never seen non-homogenous chains at class, I doubt whether this is actually a Markov Chain.
Thanks in advance
Best Answer
Homogenous means that the transition probability from state $i $ to state $j $ does not change over time, that is, it remains constant. Usually, one learns about homogenous chains first, inhomogenous markov chains are often not covered in introductory courses.
In your case, from $t$ to $t+1$, the probability of $X_{t+1}=x$ is completely determined by our knowledge of $X_{t}=y$, since $X_{t+1}=y$ or $X_{t+1}=t$ (if a six was thrown at time $t $, and this will depend only on the probability of getting a 'six' (at the $t$-th throw of the die), which we are assuming constant over time.