Stochastic

Let xn be a realization of the markov chain starting

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

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

Unformatted text preview: om walk of a knight on a chess board. A chess board is an 8 by 8 grid of squares. A knight moves by walking two steps in one direction and then one step in a perpendicular direction. • • • • ⇥ • • • • By patiently examining all of the possibilities, one sees that the degrees of the vertices are given by the following table. Lines have been drawn to make the symmetries more apparent. 2 3 4 4 4 4 3 2 3 4 6 6 6 6 4 3 4 6 8 8 8 8 6 4 4 6 8 8 8 8 6 4 4 6 8 8 8 8 6 4 4 6 8 8 8 8 6 4 3 4 6 6 6 6 4 3 2 3 4 4 4 4 3 2 The sum of the degrees is 4 · 2 + 8 · 3 + 20 · 4 + 16 · 6 + 16 · 8 = 336, so the stationary probabilities are the degrees divided by 336. This problem is boring for a rook which has 14 possible moves from any square and hence a uniform stationary distribution. In exercises at the end of the chapter, we will consider the other three interesting examples: king, bishop, and queen. 1.6.3 Reversibility Let p(i, j ) be a transition probability with stationary distribut...
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