Omar Khalil - Queen's University

are described by a transition function
with iterates
P : X × F → [0, 1]
(x, A) ↦→ P (x, A)
P n (x, A) = Px(Φn ∈ A), n ∈ Z+, x ∈ X, A ∈ F
We say Φ is ϕ-irreducible if there exists a finite measure ϕ such that for all x ∈ X,
ϕ(A) > 0 ⇒ �
P n (x, A) > 0
n
We will assume for the remainder of our discussions that Φ is ϕ-irreducible. Next we
introduce some minimal definitions of stability concepts.
For A ∈ F define
τA � inf{k ≥ 1 : Φk ∈ A}
The chain Φ is Harris recurrent if for all x ∈ X
ϕ(A) > 0 ⇒ Px(τA < ∞) = 1
A σ-finite measure π on F is called invariant if for all A ∈ F,
�
π(A) =
P (x, A)π(dx)
If Φ is Harris recurrent then a unique invariant measure π exists. If it is finite we say Φ is
positive Harris recurrent. For an aperiodic chain the existence of such a finite π is equivalent
to the condition that
lim
n→∞ ||P n (x, ·) − π|| = 0
for all x ∈ X. We say the chain is geometrically ergodic if this convergence occurs geometri-
cally quickly ; i.e, if there exists a function M and a ρ < 1 such that
||P n (x, ·) − π|| ≤ M(x)ρ n
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