Let P be the transition probability matrix of a Markov chain. Argue that if for some positive integer r, 𝐏^r has all positive entries, then so does 𝐏^n, for all integers n ⩾ r.

Unused symbols. Show that the capacity of the channel with probability transition matrix Py |x=[ (2)/(3) (1)/(3) 0 (1)/(3) (1)/(3) (1)/(3) 0 (1)/(3) (2)/(3)] is achieved by a distribution that places zero probability on one of input symbols. What is the capacity of this channel? Give an intuitive reason why that letter is not used. | Numerade

Let P^(1) and P^(2) denote transition probability matrices for ergodic Markov chains having the same state space. Let π^1 and π^2 denote the stationary (limiting) probability vectors for the two chains. Consider a process defined as follows: (i) X0=1. A coin is then flipped and if it comes up heads, then the remaining states X1, …are obtained from the transition probability matrix P^(1) and if tails from the matrix P^(2). Is {Xn, n ⩾0} a Markov chain? If p=P{ coin comes up heads }, what is limn