Stochastic

# Let xn be a realization of the markov chain starting

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

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