20 x eyk x y f y x using the law of large numbers for

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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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