F x x n m1 x note that theorems 121 and 123 do not

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Unformatted text preview: in each state.” Theorem 1.21. Asymptotic frequency. Suppose I and R. If Nn (y ) be the number of visits to y up to time n, then Nn (y ) 1 ! n Ey T y We will see later that we may have Ey Ty = 1 in which case the limit is 0. As a corollary we get the following. Theorem 1.22. If I and S hold, then ⇡ (y ) = 1/Ey Ty and hence the stationary distribution is unique. In the next two examples we will be interested in the long run cost associated with a Markov chain. For this, we will need the following extension of Theorem 1.21. (Take f (x) = 1 if x = y and 0 otherwise to recover the previous result.) P Theorem 1.23. Suppose I , S , and x |f (x)|⇡ (x) < 1 then n X 1X f (Xm ) ! f (x)⇡ (x) n m=1 x Note that Theorems 1.21 and 1.23 do not require aperiodicity. To illustrate the use of Theorem 1.23, we consider 27 1.5. LIMIT BEHAVIOR Example 1.23. Repair chain (continuation of 1.7). A machine has three critical parts that are subject to failure, but can function as long as two of these parts a...
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