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6.262.Lec12

# 6.262.Lec12 - DISCRETE STOCHASTIC PROCESSES Lecture 12...

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Lecture 12 - 3/12/2010 1 DISCRETE STOCHASTIC PROCESSES Lecture 12 Renewal Processes - Chapter 3 Review: Stopping Times Wald's Theorem The Stopping Time (N(t) +1) A Lower Bound on E[N(t)] The Renewal Equation for N(t) An Approach to the Renewal Equation Using Laplace Transforms The Elementary Renewal Theorem Blackwell's Theorem Using Renewal Processes to Think about Finite Markov Chains

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Lecture 12 - 3/12/2010 2 Stopping Times Stopping time J is random, and can depend on observations up to and including X N . If experiment has proceeded to X n , the decision to stop immediately following n (i.e., to choose J = n) should be independent of X n + 1 , X n + 2 ,... . Therefore, for any fixed t, J =N(t) is not a valid stopping time, since N(t) = n means that X 1 + X 2 + - - - + X n t and X 1 + X 2 + - - - + X n + X n+1 > t, which depends on X n+1. But , for any fixed t, J =N(t) + 1 is a valid stopping time, since N(t) + 1 = n means that N(t) = n -1, and therefore X 1 + X 2 + - - - + X n-1 t and X 1 + X 2 + - - - + X n > t, which depends only on X 1, X 2, - - - , X n.