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Unformatted text preview: h (km and hence pnh (i, j ) > 0. >0 1 , km ) > 0 for 1 m n 128 CHAPTER 4. CONTINUOUS TIME MARKOV CHAINS In discrete time a stationary distribution is a solution of ⇡ p = ⇡ . Since there is no first t > 0, in continuous time we need the stronger notion: ⇡ is said to be a stationary distribution if ⇡ pt = ⇡ for all t > 0. The last condition is di cult to check since it involves all of the pt , and as we have seen in the previous section, the pt are not easy to compute. The next result solves these problems by giving a test for stationarity in terms of the basic data used to describe the chain, the matrix of transition rates ( q (i, j ) j 6= i Q(i, j ) = j=i i where i = P j 6=i q (i, j ) is the total rate of transitions out of i. Lemma 4.3. ⇡ is a stationary distribution if and only if ⇡ Q = 0. Why is this true? Filling in the definition of Q and rearranging, the condition ⇡ Q = 0 becomes X ⇡ (k )q (k, j ) = ⇡ (j ) j k6=j If we think of ⇡ (k ) as the amount of sand at k , the righ...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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