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m507b-hw5soln-s08

# m507b-hw5soln-s08 - MATH 507b ASSIGNMENT 5 SOLUTIONS SPRING...

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MATH 507b ASSIGNMENT 5 SOLUTIONS SPRING 2008 Prof. Alexander Chapter 5: (4.3) The stationary measure is unique up to constant multiples, which means the ratio of the stationary measure at any two points y and z doesn’t depend on the base point, say y or some other x . That is, μ x ( z ) μ x ( y ) = μ y ( z ) μ y ( y ) = μ y ( z ) , or equivalently, μ x ( z ) = μ x ( y ) μ y ( z ). (Note that the chain being irreducible means that none of these quantities is 0.) (4.4) μ ( x ) = 1 for all x defines one stationary measure, and the measure μ 0 corresponding to the base point 0 is another. Uniqueness up to constant multiples means 1 = μ ( k ) μ (0) = μ 0 ( k ) μ 0 (0) = μ 0 ( k ) , for all k. μ 0 ( k ) is exactly the expected number of visits to k between visits to 0. (4.8) Irreducibility means there exits a path γ from x to y , say x x 1 → · · · → x n - 1 y , with all transitions having positive probability, and all x i 6 = x . Let A be the event that this path is followed. Then > E x T x E x ( T x 1 A ) = E x ( T x | A ) P x ( A ) , and P x ( A ) > 0, so E X ( T x | A ) < . Hence by the Markov property, > E x ( T x | A ) = E x ( T x | T x > n, X n = y ) = E y ( n + T x ) = n + E y T x . Thus E y T x < .

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