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Unformatted text preview: thand side represents
the rate at which sand leaves j , while the left gives the rate at which sand
arrives at j . Thus, ⇡ will be a stationary distribution if for each j the ﬂow of
sand in to j is equal to the ﬂow out of j . More details. If ⇡ pt = ⇡ then
X
X
X
d
⇡ pt =
⇡ (i)p0 (i, j ) =
⇡ (i)
pt (i, k )Q(k, j )
t
dt
i
i
k
X
XX
=
⇡ (i)pt (i, k )Q(k, j ) =
⇡ (k )Q(k, j ) 0= k i k Conversely if ⇡ Q = 0
d
dt X ! ⇡ (i)pt (i, j ) i = X ⇡ (i)p0 (i, j ) =
t i = XX
k X
i ⇡ (i) X Q(i, k )pt (k, j ) k ⇡ (i)Q(i, k )pt (k, j ) = 0 i Since the derivative is 0, ⇡ pt is constant and must always be equal to ⇡ its value
at 0.
Lemma 4.2 implies that for any h > 0, ph is irreducible and aperiodic, so
by Theorem 1.19
lim pnh (i, j ) = ⇡ (j ).
From this we get n!1 Theorem 4.4. If a continuoustime Markov chain Xt is irreducible and has a
stationary distribution ⇡ , then
lim pt (i, j ) = ⇡ (j ) t!1 We will now consider some examples. 129 4.3. LIMITING BEHAVIOR Example 4.10. L.A. weather chain. There are three states: 1 = sunny, 2
= smoggy, 3 = rainy. The weather stays sunny for an exponentially distributed
number of days with mean 3, then bec...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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