beamer - Vrije Universiteit Amsterdam

Probabilistic Averages I
.
Theorem 3.3.1 (first part)
.
For all states j ∈ I
.
1
lim
n→∞ n
n∑
k=1
p (k)
jj
= πj
Proof for recurrent j: apply bounded convergence (see book p. 439)
1
lim
n→∞ n
n∑
k=1
1
= lim
n→∞ n
!
=
[
lim
n→∞
p (k)
jj
n∑
k=1
1
n
1
= lim
n→∞ n
n∑
(Xk = j|X0 = j)
k=1
[{Xk = j|X0 = j}] = lim
n→∞
n∑
]
{Xk = j|X0 = j}
k=1
Proof for transient j: limn→∞ p (n)
jj
[
1
n
n∑
]
{Xk = j|X0 = j}
k=1
∑ 1 n
= 0 ⇒ limn→∞
n k=1 p(k) jj = 0.
c⃝ Ad Ridder (VU) SOR– Fall 2012 14 / 36