Please answer the following question by your own work, note that the answers from other sources are not correct, thank you

Problem 2. Let (Xn)", be a Markov chain on a finite state space E with transition matrix II : Ex E -> [0, 1]. Suppose that there exists a k ( N such that II" (x, y) > 0 for all z, y e E. For ne ZA set Yn := (X,, Xnti). (a) (8p) Show that (Y,. )", is a Markov chain on E x E, and determine its transition matrix. (b) (12p) Does the distribution of Y, have a limit as n -> co? If so, determine it