hw 2 - X n is recurrent(c Assuming recurrence find the...

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205B homework, week 3; due Thursday February 12 Durrett Chapter 5 Exercises 4.3, 4.4, 4.8, 4.10 1. Give an example to show that, if X n is a Markov chain, then f ( X n ) need not be a Markov chain. 2 . Let A and B be disjoint subsets of a finite state space S . Let f ( i ) = P i ( τ A < τ B ). (a) Write down equations satisfied by ( f ( i ) : i S ). (b) Under what conditions is ( f ( i )) the unique solution of these equations? [Hint: consider the decomposition into strongly connected components] 3. Show that for K < and a > 0 there is a constant C ( K,a ) < such that: in every finite irreducible chain max i,j E i T j C ( K,a ) where K = number of states and a = min { P ( i,j ) : P ( i,j ) > 0 } . 4. Let X n be an irreducible chain with transition matrix P . Let Y n be the jump chain Y n = X ( T n ) where T 0 = 0 and T n +1 = min { m > T n : X m 6 = X ( T n ) } . (a) Show that Y n is Markov, and write its transition matrix Q in terms of P . (b) Show that ( Y n ) is recurrent iff (
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Unformatted text preview: X n ) is recurrent. (c) Assuming recurrence, find the relation between the P-invariant measure and the Q-invariant measure. (d) Deduce that, on an infinite state space, it is possible for ( Y n ) to be positive-recurrent while ( X n ) is not. 5. Let X n be an finite irreducible chain with transition matrix P . Fix a subset A of S . Define a transition matrix Q on A by q ( i,j ) = p ( i,j ) / X k ∈ A p ( i,k ) . Suppose Q is irreducible. In the case where P is reversible , find a simple explicit formula for the stationary distribution π * of Q in terms of P and its stationary distribution π . Give an example to show that the formula may not hold in the non-reversible case. 2...
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