6.262.Lec23

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

This preview shows pages 1–6. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 =>
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 ) .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 15

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online