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.
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Lecture 12 - 3/12/2010 3 Wald's Equality Theorem: Wald's equality: Let N be a stopping rule for the IID rv's X i ; i 1 { } and let S n = X 1 + ... + X n . Then ES N [ ] = EX [ ] EN [ ] . For any sample point w , S N w ( ) = X 1 w ( ) + + X Nw () w ( ) , so S N is a random variable. Proof: Let I n be a set of indicator rv's for the set of events Nn kp . (Thus I n = 1 you stop immediately following time n or later.) I n is independent of X n , X n + 1 ,... . NI S X I n n n n ωω ω af afaf == = = 11 ; E S E X I E X E IE X E X E N n n n n nn n = = = = = 1 This proof is so brief and sooo slick ! Do you get it?
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Lecture 12 - 3/12/2010 4 () {} ( ) { } ( ) { } 1 11 1 { } { } mm Nt m m m S t S t += = = − ∩ ≤ ∩ > ()1 [ ] X [ () 1 ] X( () + 1 ) [] m(t) 1 > 1, t > 0. XX m(t) 1 1 , t > 0. t X ES ENt mt t t + + =+ = =− ≥− The Stopping Time N(t) +1 S S 1 St N(t) 0 N(t) N(t)+1 N(t)+1 X X 1 2 In any renewal process, J = N(t) +1 is a random variable that satisfies the definition of a stopping time. It is the number of the first arrival after time t. (It's kind of a "next goal wins" or "sudden death" idea – stop at the first n where the sum of your winnings exceeds t.) It is a stopping rule for X i ; i 1 { } since for any m 1 the indicator random variable 1 = =⇔ Jm I is a function only of X 1 , X 2 , --- , X m. Lemma: Let m(t) = E[N(t)]. Then 1 1 > 0. t tt X ≥−∀ Proof : Since N(t) + 1 is a stopping rule, by Wald’s equality,
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Lecture 12 - 3/12/2010 5 0 () ( ) ( ) t XX x mt F x mt xF x = =+− The Renewal Equation : Laplace Transform Approach to Solution of Renewal Equation If X has a density f X (x) with a Laplace transform: then we can take the Laplace transform of both sides of the renewal equation to find the Laplace transform L m (s) of m(t). Since L(F X (x)) = L(f X (x)) /s and L(m(t) f X (t)) = L m (s)L X (s), we find that i.e., 0 L ( ) = f ( ) , Re(s) 0, sx sx e d x () () X mm X Ls L s s =+ (1 ( )) X m X s =
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Lecture 12 - 3/12/2010 6 () (1 ( )) X m X Ls s = Analysis when L m (s) is a Rational Function 2 2 0 if X dist. exponential 1 m(t) ( ) ( 1) ( ), 2( ) where R(t) 0.
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This note was uploaded on 10/21/2010 for the course EE 5581 taught by Professor Moon,j during the Spring '08 term at Minnesota.

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

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