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Unformatted text preview: that there are about n/Ey Ty returns
by time n.
Proof. We have already shown (1.15). To turn this into the desired result, we
note that from the deﬁnition of R(k ) it follows that R(Nn (y )) n < R(Nn (y )+
1). Dividing everything by Nn (y ) and then multiplying and dividing on the end
by Nn (y ) + 1, we have
n
R(Nn (y ) + 1) Nn (y ) + 1
R(Nn (y )) <
·
Nn (y )
Nn (y )
Nn (y ) + 1
Nn (y )
Letting n ! 1, we have n/Nn (y ) trapped between two things that converge to
Ey Ty , so
n
! Ey T y
Nn (y )
and we have proved the desired result. 43 1.7. PROOFS OF THE MAIN THEOREMS* Theorem 1.22. If p is an irreducible and has stationary distribution ⇡ , then
⇡ (y ) = 1/Ey Ty
Proof. Suppose X0 has distribution ⇡ . From Theorem 1.21 it follows that
Nn (y )
1
!
n
Ey T y
Taking expected value and using the fact that Nn (y ) n, it can be shown that
this implies
E⇡ Nn (y )
1
!
n
Ey T y
but since ⇡ is a stationary distribution E⇡ Nn (y ) = n⇡ (y ).
Theorem 1.23. Suppose p is irreducible, has stationary distribution ⇡ , and
P
x f (x)⇡ (x) < 1 then
n
X
1X...
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 Spring '10
 DURRETT
 The Land

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