Stochastic

# Ta minn 1 xn a and va minn 0 xn a 51 brothersister

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 < 1|Rk < 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...
View Full Document

## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

Ask a homework question - tutors are online