Let $X_n$ be a markov chain with $d \geq 3$ states and transition matrix $(p_{ij})$. Let $Y_n$ be a process that recieves values between 1 and d. it's starting distribution fulfills: $P(Y_0= d) = 0$. in addition: $$P(Y_n = i_n | Y_0 = i_0, \dots Y_{n-1} = i_{n-1}) = \begin{cases} p_{i_{n-1}i_{n}} & i_{n-1}<d, \quad n \ge 1 \\ 1_{i_n = i_{n-2}} & i_{n-1} =d, \quad n \ge 2 \end{cases}$$ Draw a sample function of $Y_n$ that includes getting to state $d$ at least twice, from at least two different states.
I'm not sure I understand what is being asked of me here, any help is supremely appreciated. Thanks!
Right so what the top brackets essentially boil down to is that Y shares the transition probabilities of $X_n$ except when $i_n =d$, in which case $Y_n$ returns to the state from which it came to $d$, $i_{n+1} = i_{n-1}$, so a valid sample function looks like:
confirmed with my TA