Omar Khalil - Queen's University

3 Summary of Results
Theorem 1. Suppose that Φ is a ϕ-irreducible chain on X and let n(x) be a measurable
function from X to Z+.
(i) The chain is Harris recurrent if there exists a petite set C and a function V ≥ 0 un-
bounded off petite sets such that
�
P n(x) (x, dy)V (y) ≤ V (x), ∀ x ∈ C c
(ii) The chain is positive Harris recurrent if there exists a petite set C and a function V ≥ 0
bounded on C as well as a positive constant b such that
in which case
�
P n(x) (x, dy)V (y) ≤ V (x) − n(x) + bIC(x), ∀ x ∈ X
Ex[τC] ≤ V (x) + b, ∀ x ∈ X
(iii) The chain is geometrically ergodic if it is aperiodic and if there exists some petite set C,
a function V ≥ 1 bounded on C and positive constants λ < 1 and b such that
�
P n(x) (x, dy)V (y) ≤ λ n(x) [V (x) + bIc(x)], ∀x ∈ X (1)
When (1) holds there exist constants r > 1 and R < ∞ such that
�
where π denotes the invariant measure.
n
r n ||P n (x, ·) − π|| ≤ RV (x), ∀x ∈ X
Next we state the main results for random state-dependent times. To analyse those the
paper considers the chain whose next position when Φ0 = x is given by Φζx where ζx is a
random variable with distribution ax on Z+. We define the transition probabilities for this
randomly sampled chain by
�
Ka(x, A) � Px(Φζx ∈ A) = ax(n)P n (x, A)
6
n