6.262.Lec23

6.262.Lec23 - DISCRETE STOCHASTIC PROCESSES Lecture 23...

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Lecture 23 4/30/2010 Discrete Stochastic Processes 1 DISCRETE STOCHASTIC PROCESSES Lecture 23 Random Walks & Martingales: Sections 7.1 - 7.6.1 Review: Chernoff Bound Optimizing the Chernoff Bound Wald’s Identity A genuine global bound for random walks Introduction to Martingales
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Lecture 23 4/30/2010 Discrete Stochastic Processes 2 Random Walks: If X 1 , X 2 , are IID random variables and S n = X 1 + X 2 + + X n ; n = 1,2, then the process S n ; n 1 { } is a random walk. Chernoff Bound: X n X ra ra -ra g() γ (r)-ra ra SX n γ (r)-r r (Markov says) (e e ) E[e ]/e = g ( )e ( ) ln(g ( )) ln(E[e ]) e g () = g(), so () ln (g ()) n (e e ) e , r convergenc X n n rX rX X r rX XX rX nn X rS Pr rr P r r r P α γ γγ ≥≤ == ±²³ X n γ (r)-r e region e , r c o n v e r gence region, r 0. n PS
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Lecture 23 4/30/2010 Discrete Stochastic Processes 3 The minimum value of the bound for α > 0 is equal the negative of the x-axis intercept of the tangent at r 0 : Note: The slope in the figure should be E[X] < 0. [n ( ) ] ( ) ( ( )) e , r in [0, r ] X rr nr nX PS g r e γ α + ≥≤ = 0 0 ' 0 () [ r ] ' 0 ( ) e , where (r ) . r r n n αγ =
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Lecture 23 4/30/2010 Discrete Stochastic Processes 4 (Note: Slope s Note the behavior of this bound as n decreases from a huge value to 1. In particular, the x-axis intercept always lies to the right of r*, i.e., 0 0 ' 0 () [ r ] ' 0 ( ) e , where (r ) , 0. r r n PS n γ α αγ ≥≤ = > 0 0 ' 0 r ] * ' 0 ( ) e e , n 1, where (r*) = 0, (r ) 0. r r r n n γγ =>
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Lecture 23 4/30/2010 Discrete Stochastic Processes 5 Threshold Crossing Theorem Suppose E[X] < 0, g X (r) = E[e rX ] converges for all r > 0 and γ (r) = ln (g X (r)). Then for all n 1 and all α > 0, 0 0 ' 0 () [ r ] *' 0 ( ) e e , where (r*) = 0, (r ) .
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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.Lec23 - DISCRETE STOCHASTIC PROCESSES Lecture 23...

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