Stochastic

# Find conditions that guarantee that a 0 is recurrent b

This preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: 1 cards is ⇡ N log N , while the variance is ⇡ N 2 k=1 1/k 2 . 75 1.12. EXERCISES 1.69. Algorthmic e ciency. The simplex method minimizes linear functions by moving between extreme points of a polyhedral region so that each transition decreases the objective function. Suppose there are n extreme points and they are numbered in increasing order of their values. Consider the Markov chain in which p(1, 1) = 1 and p(i, j ) = 1/i 1 for j < i. In words, when we leave j we are equally likely to go to any of the extreme points with better value. (a) Use (1.25) to show that for i > 1 Ei T1 = 1 + 1/2 + · · · + 1/(i 1) (b) Let Ij = 1 if the chain visits j on the way from n to 1. Show that for j < n P (Ij = 1|Ij +1 , . . . In ) = 1/j to get another proof of the result and conclude that I1 , . . . In 1 are independent. Inﬁnite State Space 1.70. General birth and death chains. The state space is {0, 1, 2, . . .} and the transition probability has p(x, x + 1) = px p(x, x 1) = qx p(x, x) = rx for x > 0 for x 0 while the other p(x, y ) = 0. Let Vy = min{n 0 : Xn = y }...
View Full Document

## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

Ask a homework question - tutors are online