Find the stationary distribution the next exercise

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Unformatted text preview: obability escapes o to in nity or ,1. How can we keep the probability from escaping? Here is an example. 1.22 Exercise. Consider a Markov chain on the integers with P i; i + 1 = :4 and P i; i , 1 = :6 for i 0; P i; i + 1 = :6 and P i; i , 1 = :4 for i 0; P 0; 1 = P 0; ,1 = 1=2: This is a chain with in nitely many states, but it has a sort of probabilistic restoring force" that always pushes back toward 0. Find the stationary distribution. The next exercise may look a bit inscrutable at rst, but it is well worth doing and it introduces an important idea. 1.23 Exercise Probability flux . Consider a partition of the state space S of a Markov chain into two subsets A and Ac . Suppose the Markov chain has stationary distribution . Show that XX XX 1.24 i2A j 2Ac iP i; j  = i2Ac j 2A iP i; j : 1.25 Exercise. Use exercise 1.23 to re-do Exercise 1.21, by writing the equations produced by 1.24 with the choice A = f0; 1; : : : ; ig for various i. The calculation should be easier. The left side of 1.24 may be thought of the probability ux owing out of A into Ac ." The equality says that this must be the same as the ux from Ac back into A. This has the suggestive interpretation that the stationary probabilities describe a stable system in which all the probability is happy where it is, and does not want to ow to anywhere else, so that the net ow from A to Ac must be zero. We can say this in a much less mysterious way as follows. Think of i as the long run fraction of time that the chain is in state i. We will soon see a theorem  a strong law of large numbers for Markov chains" that supports this interpretation. Then iP i; j  is the long run fraction of times that a transition from i to j takes place. But clearly the long run fraction of times occupied by transitions going from a state in A to a state in Ac must equal the long run fraction of times occupied by transitions going the opposite way. In fact, along any sample path, the numbers of Stochastic Processes J. Chang, March 30, 1999 1.6. IRREDUCIBILITY, PERIODICITY, AND RECURRENCE Page 1-13 transitions that have occurred in the two directions up to any time n may di er by at most 1! 1.26 Exercise Renewal theory, the residual, and length-biased sampling . Let X1 ; X2 ; : : : be iid taking values in f1; : : : ; dg. These are typically thought of as lifetimes of lightbulbs. . . De ne Sk = X1 +    + Xk , n = inf fk : Sk  ng, and Rn = S n , n. Then Rn is called the residual lifetime at time n. This is the amount of lifetime remaining in the bulb that is in operation at time n. 1. The sequence R0 ; R1 ; : : : is a Markov chain. What is its transition matrix? What is the stationary distribution? 2. De ne the total lifetime Ln at time n by Ln = X n . This has an obvious interpretation as the total lifetime of the lightbulb in operation at time n. Show that L0 ; L1 ; : : : is not a Markov chain. But Ln still has a limiting distribution, and we'd like to nd it. We'll do this by constructing a Markov c...
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