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# Lec18 - Stat 150 Stochastic Processes Spring 2009 Lecture...

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Stat 150 Stochastic Processes Spring 2009 Lecture 18: Markov Chains: Examples Lecturer: Jim Pitman A nice collection of random walks on graphs is derived from random movement of a chess piece on a chess board. The state space of each walk is the set of 8 × 8 = 64 squares on the board: a b c d e f g h 1 2 3 4 5 6 7 8 Each kind of chess piece at a state i on an otherwise empty chess board has some set of all states j to which it can allowably move. These states j are the neighbours of i in a graph whose vertices are the 64 squares of the board. Ignoring pawns, for each of king, queen, rook, bishop and knight, if j can be reached in one step from i , this move can be reversed to reach i from j . Note the pattern of black and white squares, which is important in the following discussion. The King interior corner edge 8 possible moves 3 possible moves 5 possible moves 1

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Lecture 18: Markov Chains: Examples 2 In the graph, i ←→ j means that j is a king’s move from i , and i is a king’s move from j . Observe that N ( i ) := #( possible moves from i ) ∈ { 3 , 5 , 8 } . From general discussion of random walk on a graph in previous lecture, the reversible equilibrium distribution is π i = N ( i ) Σ where Σ := X j N ( j ) = (6 × 6) × 8 + (4 × 6) × 5 + 4 × 3 Question : Is this walk regular? Is it true that m : P m ( i, j ) > 0 for all i, j ?
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Lec18 - Stat 150 Stochastic Processes Spring 2009 Lecture...

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