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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 )
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.)
Theorem 1.23. Suppose I , S , and x |f (x)|⇡ (x) < 1 then
f (Xm ) !
f (x)⇡ (x)
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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