beamer - Vrije Universiteit Amsterdam

Equilibrium
Suppose that the Markov chain satisfies the unichain condition with a finite
set of transient states (|T| < ∞). For all states j ∈ I (and arbitrary r ∈ I):
(πP)j = ∑
πipij = ∑ (
!
= lim
n→∞
i∈I
∑
i∈I
1
= lim
n→∞ n
= lim
n→∞
1
n
n∑
n∑
k=1
i∈I
p
k=1
(k+1)
rj
1
( ∑n+1
p
n
k=1
(k)
rj − prj
(
n + 1
= lim
n→∞ n
lim
n→∞
1
n
p (k)
ri pij
1
= lim
n→∞ n
n∑
k=1
1 ∑n+1
= lim
n→∞ n
)
1 ∑n+1
n + 1
k=1
p (k)
rj
k=2
p (k)
)
ri pij
n∑ ∑
k=1
p (k)
rj
i∈I
1
−
n prj
)
= πj
p (k)
ri pij
c⃝ Ad Ridder (VU) SOR– Fall 2012 22 / 36