beamer - Vrije Universiteit Amsterdam

Proper First-Passage Times
.
Corollary
.
fij = 1 if either
(i). i, j ∈ R the same recurrent irreducible set;
(ii). i ∈ T transient, j ∈ R recurrent irreducible set; and fiR = 1.
.
.
Theorem 3.5.7(a)
.
Suppose that the unichain condition holds (see lecture 3) with |T| < ∞; then
. fij = 1 for all i ∈ I and j ∈ R.
Proof for i ∈ T:
1 − fiR = (chain nevers reaches R|X0 = 1)
= ( ∩∞
)
{Xk ∈ T} X0 = i = lim
n→∞ P
( ∩n
)
{Xk ∈ T} X0 = i
k=1
= lim
n→∞ P(Xn ∈ T|X0 = i) = lim
n→∞
∑
j∈T
k=1
p (n)
ij
finite sum
=
∑
j∈T
lim
n→∞ p(n) ij
= 0.
c⃝ Ad Ridder (VU) SOR– Fall 2012 7 / 36