Unformatted text preview: esult follows.
Proof of Theorem 5.17. Let a < b, let S0 = 0 and deﬁne stopping times for
k 1 by
Rk = min{n Sk Sk = min{n 1 : Xn a} Rk : X n b} Using the reasoning that led to Lemma 5.18
P (Sk < 1Rk < 1) a/b
Iterating we see that P (Sk < 1) (a/b)k . Since this tends to 0 as k ! 1
Xn crosses from below a to above b only ﬁnitely many times. To conclude from
this that limn!1 Xn exists, let
Y = lim inf Xn
n!1 and Z = lim sup Xn .
n!1 If P (Y < Z ) > 0 then there are numbers a < b so that P (Y < a < b < Z ) > 0
but in this case Xn crosses from below a to above b inﬁnitely many times with
positive probability, a contradiction.
To prove EX1 EX0 note that for any time n and positive real number
M
EX0 = EXn E (Xn ^ M ) ! E (X1 ^ M )
where the last conclusion follows from the reasoning in the proof of Theorem
5.14. The last conclusion implies EX0 E (X1 ^ M ) " EX1 as M " 1. Example 5.14. Polya’s urn. Consider an urn that contains red and green
balls. At time 0 there are k balls with at least one b...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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