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Essentials of Stochastic Processes - Duke University

IEssentials of Stochastic ProcessesRick Durrett304050607010 Sep10 Jun10 Mayat expiry0102050052054056058060062064066068 0700 Almost Final Version of the 2nd Edition, December, 2011 Copyright 2011, All rights the first undergraduate course in probability and the first graduatecourse that uses measure theory, there are a number of courses that teachStochastic Processesto students with many different interests and with varyingdegrees of mathematical sophistication. To allow readers (and instructors) tochoose their own level of detail, many of the proofs begin with a nonrigorousanswer to the question Why is this true? followed by aProofthat fills inthe missing details.

To check this for the gambler’s ruin chain, we note that if you are still playing at time n, i.e., your fortune X n = i with 0 < i < N, then for any possible history of your wealth i n−1,i n−2,...i 1,i 0 P(X n+1 = i+1|X n = i,X n−1 = i n−1,...X 0 = i 0) = 0.4 since to increase your wealth by one unit you have to win your next bet. Here

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