Transition matrix of an ergodic markov chain

Reference no: EM13983585

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.

Reference no: EM13983585

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