beamer - Vrije Universiteit Amsterdam
beamer - Vrije Universiteit Amsterdam
beamer - Vrije Universiteit Amsterdam
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Probabilistic Averages II<br />
.<br />
Theorem 3.3.1 (second part)<br />
.<br />
For all states i, j ∈ I<br />
.<br />
Proof: apply (3.2.12):<br />
1<br />
n<br />
Take n → ∞.<br />
n∑<br />
k=1<br />
p (k)<br />
ij<br />
( ∑n<br />
=<br />
ℓ=1<br />
= 1<br />
n<br />
1<br />
lim<br />
n→∞ n<br />
n∑<br />
k∑<br />
k=1 ℓ=1<br />
f (ℓ)<br />
)(<br />
n − ℓ<br />
ij<br />
n<br />
n∑<br />
k=1<br />
p (k)<br />
ij<br />
= fijπj<br />
f (ℓ)<br />
ij p(k−ℓ) jj = 1<br />
n<br />
1 ∑n−ℓ<br />
n − ℓ<br />
k=0<br />
p (k)<br />
)<br />
jj<br />
n∑<br />
ℓ=1<br />
f (ℓ)<br />
ij<br />
n∑<br />
k=ℓ<br />
p (k−ℓ)<br />
jj<br />
c⃝ Ad Ridder (VU) SOR– Fall 2012 15 / 36
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