beamer - Vrije Universiteit Amsterdam

Finite Irreducible Sets
.
Corollary
.
.Finite irreducible sets consist of positive recurrent states only.
Proof: Let C be a finite irreducible set. That is,
▶ For any i ∈ C it holds that ∑
j∈C pij = 1; thus, also for all n, ∑
j∈C p(n) ij = 1;
▶ All states in C communicate;
▶ All states have the same classification;
Suppose transient or nulrecurrent, then all πj = 0. Gives a contradiction:
0 = ∑
πj = ∑ (
j∈I
j∈I
finite set ∑ 1
= lim
n→∞ n
j∈I
lim
n→∞
n∑
k=1
1
n
n∑
k=1
p (k)
ij
p (k)
)
ij
1
= lim
n→∞ n
n∑ ∑
c⃝ Ad Ridder (VU) SOR– Fall 2012 20 / 36
k=1
j∈I
p (k)
ij
= 1