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Unformatted text preview: x (y )/Ex Tx
is a stationary distribution. Since µx (x) = 1 we see that
⇡ (x) = 1
Ex T x If there are k states then the stationary distribution ⇡ can be computed by
the following procedure. Form a matrix A by taking the ﬁrst k 1 columns of
p I and adding a ﬁnal column of 1’s. The equations ⇡ p = ⇡ and ⇡1 + · · · ⇡k = 1
are equivalent to
⇡A = 0 . . . 0 1
so we have ⇡= 0 ... 0 1A 1 or ⇡ is the bottom row of A 1 .
In two situations, the stationary distribution is easy to compute. (i) If
P
the chain is doubly stochastic, i.e., x p(x, y ) = 1, and has k states, then the
stationary distribution is uniform ⇡ (x) = 1/k . (ii) ⇡ is a stationary distribution
if the detailed balance condition holds
⇡ (x)p(x, y ) = ⇡ (y )p(y, x)
Birth and death chains, deﬁned by the condition that p(x, y ) = 0 if x y  > 1
always have stationary distributions with this property. If the state space is
`, ` + 1, . . . r then ⇡ can be found by setting ⇡ (`) = c, solving for ⇡ (x) for
` < x r, and then choosing c to make the probabilities sum to 1.
Convergence theorems Transient states y have pn (x, y ) !...
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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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