beamer - Vrije Universiteit Amsterdam

Observations
We might conclude
1. limn→∞ p (n)
ij
= 0 for transient j (and all i);
2. limn→∞ p (n)
ij = πj for i, j in the same aperiodic irreducible recurrent subset
R;
3. limn→∞ 1
n
4. Where ∑
j∈R πj = 1;
5. limn→∞ p (n)
ij
6. limn→∞ 1
n
∑n k=1 p(k) ij = πj for i, j in the same irreducible recurrent subset R;
= fiRπj for j ∈ R an aperiodic irreducible recurrent subset, and
i ̸∈ R (for instance transient);
∑n k=1 p(k) ij = fiRπj for j ∈ R an irreducible recurrent subset, and
i ̸∈ R (for instance transient);
Property 1 is known from previous lecture; proofs (partly) of 2-6 on the
following slides.
c⃝ Ad Ridder (VU) SOR– Fall 2012 11 / 36