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On normal approximation for φ-mixing and m-dependent random variables
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Journal Article
On normal approximation for φ-mixing and m-dependent random variables
2023
Overview
In this paper, we estimate the difference |
E
h
(
Z
n
)
−
E
h
(
Y
)
|
between the expectations of real finite Lipschitz function
h
of the sum
Z
n
= (
X
1
+ ⋯ +
X
n
)
/B
n
, where
B
n
2
=
E
(
X
1
+ ⋯ +
X
n
)
2
>
0, and a standard normal random variable
Y
, where real centered random variables
X
1
,X
2
,
… satisfy the
φ
-mixing condition, defined between the “past” and “ future”, or are
m
-dependent. In particular cases, under the condition
∑
r
=
1
∞
r
φ
(
r
)
<
∞
or
∑
r
=
1
∞
r
φ
1
/
2
(
r
)
<
∞
, the obtained upper bounds for
φ
-mixing random variables are of order
O
(
n
−
1
/
2
). In addition, we refine the previously known upper bounds of order
O
((
m
+ 1)
1+
δ
L
2+
δ,n
), where
L
2+
δ,n
is the Lyapunov fraction of order 2 +
δ
, for
m
-dependent random variables, supplementing them with explicit constants. We also separately present the case of independent r.v.s.
Publisher
Springer US,Springer Nature B.V
