Lecture Notes - Department of Mathematics and Statistics - Queen's ...

44CHAPTER 4. MARTINGALES AND FOSTER-LYAPUNOV CRITERIA FOR STABILIZATION OF MARKOV CHAINS
4.3.1 Lyapunov conditions: Geometric ergodicity
Roberts and Rosenthal use this approach to establish geometric ergodicity.
An irreducible Markov chain satisfies the univariate drift condition if there are constants λ ∈ (0, 1) and b < ∞,
along with a function W : X → [1, ∞), and a small set C such that
PV ≤ λV + b1 C . (4.12)
One can now define a bivariate drift condition for two independent copies of a Markov chain with a small set C.
This condition requires that there exists a function h : X × X → [1, ∞) and α > 1 such that
where
¯Ph(x, y) ≤h(x, y)/α (x, y) /∈ C × C
¯Ph(x, y) b
1−λ
− 1, then the
bivariate drift condition is satisfied for h(x, y) = 1 2 (V (x) + V (y)) and α−1 = λ + b/(d + 1) > 1.
The bivariate condition ensures that the processes x, x ′ hits the set C × C, from where the two chains may be
coupled. The coupling inequality then leads to the desired conclusion.
Theorem 4.3.3 (Theorem 9 of [28]) Suppose {x t } is an aperiodic, irreducible Markov chain with invariant
distribution π(·). Suppose C is a (1, ǫ, ν)-small set and V :→ [1, ∞) satisfies the univariate drift condition (4.12)
with constants λ ∈ (0, 1) and b < ∞ with V (x) < ∞ for some x ∈ X. Then {x t } is geometrically ergodic.
As a side remark, we note the following observation. If the univariate drift condition holds then the sequence of
random variables M n = λ −n V (x n ) − n−1 ∑
b1 C (x k ) is a supermartingale and thus we have the inequality
k=0
for all n and x 0 ∈ X, and with Theorem 3.3.4 we also have
for all B ∈ B + (X).
n−1
∑
E x0 [λ −n V (x n )] ≤ V (x 0 ) + E x0 [ b1 C (x k )] (4.13)
k=0
E x [λ −τB ] ≤ V (x) + c(B)
4.3.2 Subgeometric ergodicity
Here, we review the class of subgeometric rate functions (see section 4 of [16], section 5 of [12], [23], [13], [30]).
Let Λ 0 be the family of functions r : N → R >0 such that
r is non-decreasing, r(1) ≥ 2
and
log r(n)
n
↓ 0
as n → ∞