A uniform Berry--Esseen theorem on M -estimators for geometrically ergodic Markov chains
Abstract
Let
{
X
n
}
n≥0
be a
V
-geometrically ergodic Markov chain. Given some real-valued functional
F
, define
M
n
(α):=
n
−1
∑
n
k=1
F(α,
X
k−1
,
X
k
)
,
α∈A⊂R
. Consider an
M
estimator
α
^
n
, that is, a measurable function of the observations satisfying
M
n
(
α
^
n
)≤
min
α∈A
M
n
(α)+
c
n
with
{
c
n
}
n≥1
some sequence of real numbers going to zero. Under some standard regularity and moment assumptions, close to those of the i.i.d. case, the estimator
α
^
n
satisfies a Berry--Esseen theorem uniformly with respect to the underlying probability distribution of the Markov chain.