Convergence of moments in a Markov-chain central limit theorem

• Published in: Indagationes mathematicae Vol. 12; no. 4; pp. 533 - 555
• Main Author: Steinsaltz, David
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 17-12-2001
• ISSN: 0019-3577; 1872-6100
• DOI: 10.1016/S0019-3577(01)80041-0

Let (
X
i
)
i
=0
∞ be a
V-uniformly ergodic Markov chain on a general state space, and let π be its stationary distribution. For
g : χ →
R, define
▪ It is shown that if |
g| ≤
V
1/
n
for a positive integer
n, then
E
x
W
k
(
g)
n
converges to the
n-th moment of a normal random variable with expectation 0 and variance
▪ This extends the existing Markov-chain central limit theorems, according to which expectations of bounded functionals of
W
k
(
g) converge.
We also derive nonasymptotic bounds for the error in approximating the moments of
W
k
(
g) by the normal moments. These yield easy bounds of all feasible polynomial orders, and exponential bounds as well under some circumstances, for the probabilities of large deviations by the empirical measure along the Markov chain path
X
i
.