Lecture 12 - 3/12/20101DISCRETESTOCHASTICPROCESSES 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 - 3/12/20102Stopping Times Stopping time Jis random, and can depend on observations up to and includingXN. If experiment has proceeded to Xn, the decision to stop immediately following n(i.e., to choose J = n) should be independent of Xn+1,Xn+2,.... Therefore, for any fixed t, J =N(t) is not a valid stopping time, since N(t) = n means that X1 + X2 + - - - + Xn≤t and X1 + X2 + - - - + Xn+ Xn+1 > t, which depends on Xn+1. But , for any fixed t, J =N(t) + 1 isa valid stopping time, since N(t) + 1 = n means that N(t) = n -1, and therefore X1 + X2 + - - - + Xn-1≤t and X1 + X2 + - - - + Xn > t, which depends only on X1,X2, - - - , Xn.