1. Let P be the transition matrix of an ergodic Markov chain. Let x be any column vector such that Px = x. Let M be the maximum value of the components of x. Assume thatxi = M . Show that if pij > 0 then xj = M . Use this to prove that x must be a constant vector.
2. Let P be the transition matrix of an ergodic Markov chain. Let w be a fixed probability vector (i.e., w is a row vector with wP = w). Show that if wi = 0 and pji > 0 then wj = 0. Use this to show that the fixed probability vector for an ergodic chain cannot have any 0 entries.
3. Find a Markov chain that is neither absorbing or ergodic.