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The random walk in one dimension is certain recurrent. at, for large n, u2n N de a proof, courtesy of Theorem2. Since is divergent, the clain

The random walk in one dimension is certain recurrent. at, for large n, u2n N de a proof, courtesy of Theorem2. Since is divergent, the clain

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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(2n 1 U2n= n 22n Theorem 3. The random walk in one dimension is certain recurrent. Proof We claim that, for large n, 2n would provide a proof, courtesy of Theorem 2. Let N be a binomial random variable with p = 22n By the Central Limit Theorem, for large n 1 P(n-≤N ≤N≤n+2) ~P( where Z is a standard normal random variable. 6 Since is divergent, the claim 1/2 and 2n trials. √n P(n-1 ≤N≤ n + 1) (13) <2<
Transcribed Image Text:(2n 1 U2n= n 22n Theorem 3. The random walk in one dimension is certain recurrent. Proof We claim that, for large n, 2n would provide a proof, courtesy of Theorem 2. Let N be a binomial random variable with p = 22n By the Central Limit Theorem, for large n 1 P(n-≤N ≤N≤n+2) ~P( where Z is a standard normal random variable. 6 Since is divergent, the claim 1/2 and 2n trials. √n P(n-1 ≤N≤ n + 1) (13) <2<
- dx ≈ πη Hence the claim is established, and the theorem follows. Exercise 2. Fill in the details concerning the application of the CLT in the proof. Now we can give a proof of Theorem, and a little more: Proof Since the movements in each dimension are assumed to be independent, by Theorem 3, when n is large 1 (14) U2n (Tn) vi-0 So i = ∞ for d = 1,2; but <∞o for d 23. By Theorem the drunkard's walk in certain recurrent in dimensions 1 and 2; but uncertain recurrent in dimensions 3 and higher. Exercise 3. Much more is known about all this. Some things for you to research
Transcribed Image Text:- dx ≈ πη Hence the claim is established, and the theorem follows. Exercise 2. Fill in the details concerning the application of the CLT in the proof. Now we can give a proof of Theorem, and a little more: Proof Since the movements in each dimension are assumed to be independent, by Theorem 3, when n is large 1 (14) U2n (Tn) vi-0 So i = ∞ for d = 1,2; but <∞o for d 23. By Theorem the drunkard's walk in certain recurrent in dimensions 1 and 2; but uncertain recurrent in dimensions 3 and higher. Exercise 3. Much more is known about all this. Some things for you to research
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