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Unformatted text preview: let Ty = min{n
1:
Xn = y } and let
⇢xy = Px (Ty < 1) When x 6= y this is the probability Xn ever visits y starting at x. When x = y
this is the probability Xn returns to y when it starts at y . We restrict to times
n 1 in the deﬁnition of Ty so that we can say: y is recurrent if ⇢yy = 1 and
transient if ⇢yy < 1.
Transient states in a ﬁnite state space can all be identiﬁed using
Theorem 1.5. If ⇢xy > 0, but ⇢yx < 1, then x is transient.
Once the transient states are removed we can use
Theorem 1.7. If C is a ﬁnite closed and irreducible set, then all states in C
are recurrent.
Here A is closed if x 2 A and y 62 A implies p(x, y ) = 0, and B is irreducible if
x, y 2 B implies ⇢xy > 0.
The keys to the proof of Theorem 1.7 are: (i) If x is recurrent and ⇢xy > 0
then y is recurrent, and (ii) In a ﬁnite closed set there has to be at least one
recurrent state. To prove these results, it was useful to know that if N (y ) is
the number of visits to y at times n 1 then
1
X pn (x, y ) = Ex N (y ) = n=1...
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 Spring '10
 DURRETT
 The Land

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