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Unformatted text preview: ishop can move any number of squares diagonally. Let Xn be the sequence of squares that results if we pick one of
bishop’s legal moves at random. Find (a) the stationary distribution and (b)
the expected number of moves to return to corner (1,1) when we start there.
1.51. Queen’s random walk. A queen can move any number of squares horizontally, vertically, or diagonally. Let Xn be the sequence of squares that results
if we pick one of queen’s legal moves at random. Find (a) the stationary distribution and (b) the expected number of moves to return to corner (1,1) when
we start there.
1.52. Wright–Fisher model. Consider the chain described in Example 1.7.
✓◆
N
p(x, y ) =
(⇢x )y (1 ⇢x )N y
y
where ⇢x = (1 u)x/N + v (N
x)/N . (a) Show that if u, v > 0, then
limn!1 pn (x, y ) = ⇡ (y ), where ⇡ is the unique stationary distribution. There is
P
no known formula for ⇡ (y ), but you can (b) compute the mean ⌫ = y y ⇡ (y ) =
limn!1 Ex Xn .
1.53. Ehrenfest chain. Consider the Ehrenfest chain, Example 1.2, with transit...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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