6.262.Lec22

6.262.Lec22 - DISCRETE STOCHASTIC PROCESSES Lecture 22...

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Lecture 22 4/28/2010 Discrete Stochastic Processes 1 DISCRETE STOCHASTIC PROCESSES Lecture 22 Random Walks: Sections 7.1 - 7.5.3 Threshold Crossing: Large Deviation Techniques Optimization of the Chernov bound Wald’s Identity Examples
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Lecture 22 4/28/2010 Discrete Stochastic Processes 2 R ANDOM W ALKS : 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. Typical questions: Find P sup n S n a ( ) (distribution of maximum). PS n a and S i < a ,1 i < n () (first crossing distribution) n b and S i < a i < n for b > a (overshoot).
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Lecture 22 4/28/2010 Discrete Stochastic Processes 3 R ANDOM W ALKS & T HRESHOLD C ROSSING L ARGE D EVIATION T ECHNIQUES The usual stopping rule is stop when the RW crosses either a large positive threshold a or a large negative threshold b . For the examples, we were interested in low probability events (waiting too long in a queue, making errors, etc.). We focus on large deviation techniques – approximating or bounding very small probabilities. These are based on moment generating functions . gr ( ) = E exp rX ( ) [ ] = exp rx ( ) dF X x ( ) The “region of convergence” of g ( r ) is the set of r for which this integral converges. Assume the region of convergence is large enough for everything we want to do.
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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.Lec22 - DISCRETE STOCHASTIC PROCESSES Lecture 22...

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