Please help on this Applied Stochastic Processesquestion

3. (14 points) Let p 6 (0,1) and q = 1 h p. Consider the stochastic matrix (133000 (103000 P2:100:00 q000p indexed by the non-negative integers S = {0, 1, 2, 3, }. In this example we will do the following. (a) Show P is irreducible and aperiodic. b Show that P is positive recurrent and nd 113- p- for every 3' E S. Hint: see lecture 17 J 3 (0) Consider watching a (stationary discrete time Markov chain) process evolve with transition matrix P given above. Approximate the probability that, after watching for some time, the process is in state 3. Does the answer depend on how the process was initially distributed over the states in S