Whether the sequence is a Markov chain

[vilgeforc5]

Let η be a Markov chain with a set of states [imath]\left \{ 1,2,3 \right \}[/imath] and a matrix of transition probabilities [imath]\begin{bmatrix} 0 & 1-a & a\\ 1-b & 0 & b\\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{bmatrix}[/imath]


Let's define a sequence [imath]\xi_n, n \geq 0[/imath] in the following way: [imath]\left\{\begin{matrix} 1 , \eta = 1,\eta =2 \\2 , \eta = 3 \end{matrix}\right. [/imath]
Under what condition is the sequence [imath]\xi_n[/imath] also a Markov chain?

 

[blamocur]

If the current probability distribution of [imath]\eta_n[/imath] is [imath]p_1,p_2,p_3[/imath] what is the probability distribution for [imath]\xi_{n+1}[/imath]?