Volume 8 (2007), Issue 4, Article 92, 10 pp.
IMPROVEMENT OF THE NON-UNIFORM VERSION OF BERRY-ESSEEN
INEQUALITY VIA PADITZ-SIGANOV THEOREMS
K. NEAMMANEE AND P. THONGTHA
D EPARTMENT OF M ATHEMATICS ,FACULTY OF S CIENCE ,
C HULALONGKORN U NIVERSITY
BANGKOK 10330, T HAILAND
Kritsana.n@chula.ac.th
Received 28 June, 2007; accepted 05 October, 2007
Communicated by T.M. Mills
A BSTRACT. We improve the constant in a non-uniform bound of the Berry-Esseen inequality
without assuming the existence of the absolute third moment by using the method obtained from
the Paditz-Siganov theorems. Our bound is better than the results of Thongtha and Neammanee
in 2007 ([14]).
Key words and phrases: Berry-Esseen inequality, Paditz-Siganov theorems, central limit theorem, uniform and non-uniform
bounds.
2000 Mathematics Subject Classification. 60F05, 60G50.
1. I NTRODUCTION AND M AIN R ESULTS
The Berry-Esseen inequality is one of the most important inequalities in the theory of probability. This inequality was independently discovered by two mathematicians, Andrew C. Berry
([2]) and Carl-Gustav Esseen ([5]) in 1941 and
P 1945 respectively. Let X1 , X2 , . . . , Xn be independent random varibles with zero mean and ni=1 EXi2 = 1. Define Wn = X1 +X2 +· · ·+Xn .
Then Var Wn = 1. Let Fn be the distribution function of Wn and Φ the standard normal distribution function, i.e.,
Z x
t2
1
Fn (x) = P (Wn ≤ x) and Φ(x) = √
e− 2 dt.
2π −∞
The central limit theorem shows that Fn converges pointwise to Φ as n → ∞ and the bounds
of this convergence are,
n
X
(1.1)
sup |P (Wn ≤ x) − Φ(x)| ≤ C0
E|Xi |3
x∈R
i=1
and
(1.2)
215-07
|P (Wn ≤ x) − Φ(x)| ≤
n
C1 X
E|Xi |3
1 + |x|3 i=1
2
K. N EAMMANEE AND P. T HONGTHA
for uniform and non-uniform versions respectively, where both C0 ans C1 are positive constants
and stated under the assumption that E|Xi |3 < ∞ for i = 1, 2, . . . , n.
In the case of identical Xi ’s, Siganov ([11]) and Chen ([5]) improved the constant down to
0.7655 and 0.7164, respectively. For non-uniform bounds, Nageav ([7]) was the first to obtain
(1.2) and Michel ([6]) calculated the constant to be 30.84.
Without assuming identically distributed Xi0 s, Beek ([15]) sharpened the constant down to
0.7975 in 1972 for the uniform version. The best bound was found by Siganov ([11]) in 1986.
Theorem 1.1 (Siganov,1986). Let X1 , X2 , . . . , Xn be independent
random variables such that
P
EXi = 0 and E|Xi |3 < ∞ for i = 1, 2, . . . , n. Assume that ni=1 EXi2 = 1. Then
sup |P (Wn ≤ x) − Φ(x)| ≤ 0.7915
x∈R
n
X
E|Xi |3 ,
i=1
where Wn = X1 + X2 + · · · + Xn .
For the non-uniform version, Bikelis ([1]) generalized (1.2) to this case and Paditz ([9])
calculated C1 to be 114.7 in 1977. He also improved his result down to 31.935 in 1989.
Theorem 1.2 (Paditz ([10]),1989). Under the assumptions of Theorem 1.1, we have
n
31.935 X
|P (Wn ≤ x) − Φ(x)| ≤
E|Xi |3 .
1 + |x|3 i=1
In 2001, Chen and Shao ([3]) gave new versions of (1.1) and (1.2) without assuming the
existence of third moments. Their results are
n
X
{E|Xi |2 I(|Xi | ≥ 1|) + E|Xi |3 I(|Xi | < 1)}
(1.3)
sup |P (Wn ≤ x) − Φ(x)| ≤ 4.1
x∈R
i=1
and
(1.4) |P (Wn ≤ x)−Φ(x)| ≤ C2
n X
EX 2 I(|Xi | ≥ 1 + |x|)
i
i=1
(1 + |x|)2
E|Xi |3 I(|Xi | < 1 + |x|)
+
(1 + |x|)3
,
where C2 is a positive constant and I(A) is an indicator random variable such that
(
1 if A is true,
I(A) =
0 otherwise.
In 2005, Neammanee ([8]) combined the concentration inequality in ([3]) with a coupling
approach to calculate the constant in (1.4), giving,
(1.5)
|P (Wn ≤ x) − Φ(x)|
n X
EXi2 I(|Xi | ≥ 1 + | x4 |) E|Xi |3 I(|Xi | < 1 + | x4 |)
≤ C3
+
,
x 2
x 3
(1
+
|
|)
(1
+
|
|)
4
4
i=1
where C3 is 21.44 for large values of x such that |x| ≥ 14.
Thongtha and Neammanee ([14]) improved the concentration inequality used in ([8]) and
gave a better constant, i.e., 9.7 for |x| ≥ 14. The method which was used in ([8]) is Stein’s
method which was first introduced by Stein ([12]) in 1972. In this work, we provide a better
constant by using Paditz-Siganov theorems. The results are as follows.
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 92, 10 pp.
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I MPROVEMENT OF THE N ON - UNIFORM V ERSION OF B ERRY-E SSEEN I NEQUALITY V IA PADITZ -S IGANOV T HEOREMS
3
Theorem 1.3. We have
|P (Wn ≤ x) − Φ(x)| ≤ C
n X
EX 2 I(|Xi | ≥ 1 + |x|)
i
(1 + |x|)2
i=1
where
C=
E|Xi |3 I(|Xi | < 1 + |x|)
+
(1 + |x|)3
,

49.89 if 0 ≤ |x| < 1.3,






59.45 if 1.3 ≤ |x| < 2,




 73.52 if 2 ≤ |x| < 3,

76.17 if 3 ≤ |x| < 7.98,






45.80 if 7.98 ≤ |x| < 14,



 39.39 if |x| ≥ 14.
To compare Theorem 1.3 with the result of Thongtha and Neammanee ([14]) in (1.5), we
give Corollary 1.4.
Corollary 1.4. We have
|P (Wn ≤ x) − Φ(x)| ≤ C
n X
EXi2 I(|Xi | ≥ 1 + | x |)
4
(1 + | x4 |)2
i=1
where
C=
E|Xi |3 I(|Xi | < 1 + | x4 |)
+
(1 + | x4 |)3
,

9.54 if 0 ≤ |x| < 1.3,






19.74 if 1.3 ≤ |x| < 2,




 18.38 if 2 ≤ |x| < 3,

14.63 if 3 ≤ |x| < 7.98,






5.13 if 7.98 ≤ |x| < 14,



 3.55 if |x| ≥ 14.
We note from Corollary 1.4 that our result is better than a bound from Thongtha and Neammanee in ([14]).
2. P ROOF OF THE M AIN R ESULTS
In this section, we will prove Theorem 1.3 by using the Paditz-Siganov theorems. Corollary
1.4 can be obtained easily from Theorem 1.3. To prove these results, let
n
X
Yi,x = Xi I(|Xi | < 1 + x),
Sx =
Yi,x ,
i=1
αx =
n
X
EXj2 I(|Xj |
≥ 1 + x),
βx =
i=1
βx
γx =
2
n
X
E|Xj |3 I(|Xj | < 1 + x),
i=1
and
αx
βx
δx =
+
for x > 0.
2
(1 + x)
(1 + x)3
Proposition 2.1. For each n ∈ N, we have
P
7αx
(1) ni=1 E|Yi,x − EYi,x |3 ≤ βx + 1+x
,
(2) 1 − 2αx ≤ Var Sx ≤ 1, and
1
(3) If αx ≤ 0.11, then 0 < √Var
≤ 1 + 1.452αx .
Sx
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 92, 10 pp.
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4
K. N EAMMANEE AND P. T HONGTHA
Proof. 1. By the fact that
|EXi I(|Xi | < 1 + x)| = |EXi I(|Xi | ≥ 1 + x)|,
n
X
2
E|Xi | ≤
EXi2 = 1 and E 2 Xi ≤ EXi2 ,
(2.1)
i=1
we have
n
X
E|Yi,x − EYi,x |3
i=1
=
n
X
E|Xi I(|Xi | < 1 + x) − EXi I(|Xi | < 1 + x)|3
i=1
≤
n
X
[E|Xi |3 I(|Xi | < 1 + x) + 3EXi2 I(|Xi | < 1 + x)|EXi I(|Xi | < 1 + x)|
i=1
+ 3E|Xi I (|Xi | < 1 + x) |E 2 Xi I (|Xi | < 1 + x) | + |EXi I (|Xi | < 1 + x) |3 ]
n
n
X
X
3
≤
E|Xi | I(|Xi | < 1 + x) + 3
|EXi I(|Xi | < 1 + x)|
i=1
i=1
+3
n
X
E|Xi ||EXi I (|Xi | < 1 + x) ||EXi I (|Xi | < 1 + x) |
i=1
+
n
X
E|Xi |2 I (|Xi | < 1 + x) |EXi I (|Xi | < 1 + x) |
i=1
≤ βx + 3
+
n
X
|EXi I(|Xi | ≥ 1 + x)| + 3
i=1
n
X
n
X
E|Xi |2 |EXi I(|Xi | ≥ 1 + x)|
i=1
|EXi I (|Xi | ≥ 1 + x) |
i=1
≤ βx + 3
n
X
E|Xi |I(|Xi | ≥ 1 + x) + 3
i=1
+
n
X
n
X
E|Xi |I(|Xi | ≥ 1 + x)
i=1
E|Xi |I (|Xi | ≥ 1 + x)
i=1
= βx + 7
≤ βx + 7
n
X
i=1
n
X
i=1
E|Xi |I(|Xi | ≥ 1 + x)
7αx
E|Xi |2 I(|Xi | ≥ 1 + x)
= βx +
.
(1 + x)
(1 + x)
2. By (2.1), we note that
Var Sx =
n
X
i=1
=
n
X
Var Yi,x =
n
X
2
(EYi,x
− E 2 Yi,x )
i=1
EXi2 I(|Xi |
< 1 + x) −
i=1
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 92, 10 pp.
n
X
E 2 Xi I(|Xi | < 1 + x)
i=1
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I MPROVEMENT OF THE N ON - UNIFORM V ERSION OF B ERRY-E SSEEN I NEQUALITY V IA PADITZ -S IGANOV T HEOREMS
=1−
n
X
EXi2 I(|Xi |
≥ 1 + x) −
i=1
E 2 Xi I(|Xi | ≥ 1 + x)
i=1
= 1 − αx −
(2.2)
n
X
5
n
X
E 2 Xi I(|Xi | ≥ 1 + x).
i=1
From this and the fact that αx ≥ 0, we have Var Sx ≤ 1.
By (2.2), we have
n
X
Var Sx = 1 − αx −
E 2 Xi I(|Xi | ≥ 1 + x)
i=1
≥ 1 − αx −
n
X
EXi2 I(|Xi | ≥ 1 + x)
i=1
= 1 − 2αx .
Hence, 1 − 2αx ≤ Var Sx ≤ 1.
3. For 0 < t ≤ 0.11, by using Taylor’s formula, we have
1
t
√
=1+
3 for some c ∈ (0, 0.11]
1 − 2t
(1 − 2c) 2
t
≤1+
3
(1 − 2(0.11)) 2
≤ 1 + 1.452t.
From this fact and 2., we have
0< √
1
1
≤√
≤ 1 + 1.452αx
1 − 2αx
Var Sx
for αx ≤ 0.11.
Yi,x −EYi,x
√
Var Sx
Pn
Proposition 2.2. For each x > 0, let Ȳi,x =
and S̄x = i=1 Ȳi,x .
(1) If αx ≤ 0.099 and 1.3 ≤ x ≤ 2, then
− ESx
x − ESx 54.513αx 41.195βx
P S¯x ≤ x
√
−Φ √
≤
+
.
(1 + x)2
(1 + x)3
Var S
Var S x
2
x
1
,
5
(2) If (1 + x) αx < then
− ESx
x − ESx C1 αx
C2 βx
P S¯x ≤ x
√
− Φ( √
≤
+
2
(1 + x)
(1 + x)3
Var Sx
Var Sx
where C1 = 57.186 C2 = 73.515 for 2 ≤ x < 3,
C1 = 33.318 C2 = 76.17 for 3 ≤ x < 7.98,
C1 = 3.976 C2 = 45.8 for 7.98 ≤ x < 14, and
C1 = 1.226 C2 = 39.382 for x ≥ 14.
Proof. 1. By Proposition 2.1(1) of ([14]) and Proposition 2.1(2), we have
αx
(2.3)
|ESx | ≤
≤ 0.043 and 1 ≥ Var Sx ≥ 0.802
1+x
which imply
x − ESx
2 + 0.043
(2.4)
0≤ √
≤ √
= 2.2813.
Var Sx
0.802
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 92, 10 pp.
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6
K. N EAMMANEE AND P. T HONGTHA
By Proposition 2.1(1) and (2.3),
n
X
3
E|Ȳi,x | =
i=1
n
X
i=1
=
Yi,x − EYi,x 3
E √
Var Sx 3
(Var Sx ) 2
1
E|Yi,x − EYi,x |3
i=1
7αx
≤
βx +
3
1+x
(Var Sx ) 2
= 1.3923βx + 4.2375αx .
(2.5)
Note that S̄x =
n
X
1
Pn
i=1
Ȳi,x is the sum of independent random variables whose
E Ȳi,x = 0 and Var S̄x = 1.
By (2.5) and Theorem 1.1,
n
X
P S̄x ≤ z − Φ(z) ≤ 0.7915
E|Ȳi,x |3
i=1
≤ 0.7915(1.3923βx + 4.2375αx )
≤ 1.102βx + 3.354αx
for all z ∈ R. From this fact, (2.3) and (2.4), we have
− ESx
x − ESx P S̄x ≤ x
√
−Φ √
Var Sx
Var Sx 3
x
1 + √x−ES
(1.102βx + 3.354αx )
Var Sx
≤
3
x−ESx
√
1+
Var Sx
≤
(3.2813)3 (1.102βx + 3.354αx )
3
x−ESx
√
1+
Var Sx
38.933βx + 118.495αx
(0.957 + x)3
41.195βx 125.379αx
≤
+
(1 + x)3
(1 + x)3
41.195βx 54.513αx
≤
+
(1 + x)3
(1 + x)2
≤
where we use the fact that
1+x
≤ 1.019 for all 1.3 < x < 2
0.957 + x
in the fourth inequality.
2. Case 2 ≤ x < 3.
We can prove the result of this case by using the same argument as 1.
Case 3 ≤ x < 7.98.
x−ESx
x−ESx √
√
To bound P S̄x ≤ Var Sx − Φ Var Sx in 1., we used Theorem 1.1.
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I MPROVEMENT OF THE N ON - UNIFORM V ERSION OF B ERRY-E SSEEN I NEQUALITY V IA PADITZ -S IGANOV T HEOREMS
7
But in this case, we will use Theorem 1.2.
We note that
0 ≤ αx ≤ 0.0125,
(2.6)
1 ≥ Var Sx ≥ 0.975,
and, by Proposition 2.1(1) of ([14]), |ESx | ≤ 0.00313.
Then, for 3 ≤ x ≤ 7.98,
1
1+
x−ESx
√
Var Sx
3 ≤ 2.29
1+
x−ESx
√
Var Sx
3
and
n
X
E|Ȳi,x |3 ≤ 1.039βx + 1.819αx .
i=1
From these facts, (2.6) and Theorem 1.2, we have
− ESx
x − ESx P S̄x ≤ x
√
−Φ √
Var Sx
Var Sx Pn
P
(31.935) i=1 E|Ȳi,x |3
(31.935)(2.29) ni=1 E|Ȳi,x |3
≤
≤
3
3
x−ESx
x
√
1 + √x−ES
1
+
Var Sx
Var Sx
(1.0008)3 (75.983βx + 132.952αx )
73.131(1.039βx + 1.818αx )
≤
(0.99687 + x)3
(1 + x)3
76.17βx
133.27αx
76.17βx
33.318αx
+
≤
+
,
≤
3
3
3
(1 + x)
(1 + x)
(1 + x)
(1 + x)2
≤
where we use the fact that
1+x
≤ 1.0008 for all 3 ≤ x < 7.98
0.99687 + x
in the fourth inequality.
Case x ≥ 7.98.
We can prove the result of this case by using the same argument as the case 3 ≤ x < 7.98.
We are now ready to prove Theorem 1.3.
Proof of Theorem 1.3. It suffices to consider only x ≥ 0 as we can simply apply the results to
−Wn when x < 0.
Case 1. 0 ≤ x < 1.3.
Note that for x ≥ 0,
EXi2 I(|Xi | ≥ 1) + E|Xi |3 I(|Xi | < 1) ≤ EXi2 I(|Xi | ≥ 1 + x) + E|Xi |3 I(|Xi | < 1 + x)
and for 0 ≤ x ≤ 1.3, (1 + x)3 ≤ 12.167.
From these facts and (1.3), we have
|P (Wn ≤ x) − Φ(x)|
n n
o
X
≤ 4.1
EXi2 I(|Xi | ≥ 1) + E|Xi |3 I(|Xi | < 1)
i=1
n n
o
X
≤ 4.1
EXi2 I(|Xi | ≥ 1 + x) + E|Xi |3 I(|Xi | < 1 + x)
i=1
n
o
4.1(12.167) X n
2
3
≤
EX
I(|X
|
≥
1
+
x)
+
E|X
|
I(|X
|
<
1
+
x)
i
i
i
i
(1 + x)3 i=1
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 92, 10 pp.
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8
K. N EAMMANEE AND P. T HONGTHA
≤ 49.89
n X
EX 2 I(|Xi | ≥ 1 + x)
i
(1 + x)2
i=1
E|Xi |3 I(|Xi | < 1 + x)
+
(1 + x)3
.
Before proving another case, we need the equation
x
−
ES
4.931αx x
−
ES
x
x
¯x ≤ √
√
(2.7)
|P (Wn ≤ x) − Φ(x)| ≤
+
P
S
−
Φ
(1 + x)2 Var Sx
Var Sx for αx ≤ 0.11 and x ≥ 1.3.
By (2.9) of ([14]), it suffices to show that for αx ≤ 0.11 and x ≥ 1.3,
x − ESx
x − ESx 3.319αx ¯
(2.8)
|P (Sx ≤ x) − Φ(x)| ≤
.
+ P Sx ≤ √
−Φ √
(1 + x)2 Var Sx
Var Sx By Proposition 2.1(1) and Proposition 2.1(2), we have
x − ESx
αx
√
≥ x − ESx ≥ x −
,
(1 + x)
Var Sx
which implies
x − ESx
αx
min x, √
≥x−
.
1+x
Var Sx
From this and the fact that
1
Φ(b) − Φ(a) = √
2π
Z
b
e
−t2
2
dt ≤ √
a
Z
1
2πe
a2
2
a
b
(b − a)
1dt = √
a2
2πe 2
for 0 < a < b, we have
(2.9)
|P (Sx ≤ x) − Φ(x)|
x
−
ES
x
−
ES
x
−
ES
x
x
x
+ Φ √
≤ P S̄x ≤ √
−Φ √
− Φ(x)
Var Sx
Var Sx
Var Sx
x − ESx
x − ESx ≤ P S̄x ≤ √
−Φ √
Var Sx
Var Sx x
1
ESx +√
−x− √
h
i2 √
x−ESx
1
Var
S
Var Sx min x, √
x
2
Var
S
x
2πe
x − ESx
x − ESx ≤ P S̄x ≤ √
−Φ √
Var Sx
Var Sx x
1
ESx √
+√
−x− √
.
αx 2 1
Var Sx
Var Sx 2πe 2 (x− 1+x )
Note that for x ≥ 1.3
x2
e 2 ≥ 0.933(1 + x),
x2
e 2 ≥ 0.193(1 + x)3
and
1
αx
2
x2
x
x2
e 2 (x− 1+x ) ≥ e 2 −( 1+x )αx ≥ 0.89e 2 .
From these facts, Proposition 2.1(1) , Proposition 2.1(3) and αx ≤ 0.11, we have
1
x
ESx √
−x− √
√
αx 2 1
Var Sx
Var Sx 2πe 2 (x− 1+x )
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≤√
1
x
+ √
√
−
x
Var S
x2
2π 0.89e 2
x
1
ESx
√
Var S
x2
2π 0.89e 2
x
9
1.452αx x
αx
(1 + 1.452αx )
√
+
2π(0.89)(0.193)(1 + x)3 (1 + x) 2π(0.89)(0.933)(1 + x)
3.373αx x 0.558αx
3.931αx
≤
+
≤
.
3
2
(1 + x)
(1 + x)
(1 + x)2
≤√
From this fact, (2.8) and (2.9), we have (2.7)
Case 2. 1.3 ≤ x < 2.
αx
By the fact that |P (Wn ≤ x) − Φ(x)| ≤ 0.55 ([3, pp. 246]), we can assume (1+x)
2 ≤ 0.011, i.e.
αx ≤ 0.099.
From this fact, (2.7) and Proposition 2.2(1), we have
4.931αx x − ESx
x − ESx |P (Wn ≤ x) − Φ(x)| ≤
+ P S̄x ≤ √
−Φ √
(1 + x)2 Var Sx
Var Sx 4.931αx
41.195βx 54.513αx
≤
+
+
2
(1 + x)
(1 + x)3
(1 + x)2
59.444αx 41.195βx
=
+
≤ 59.444δx .
(1 + x)2
(1 + x)3
Case 3. 2 ≤ x ≤ 14.
Subcase 3.1. (1 + x)2 αx ≥ 15 .
Using the same argument of subcase 1.1 in Theorem 1.2 of ([14]) and the facts that
(2.10)
x2
1+x
1
= 1 + ≤ 1.5 and e 2 ≥ 0.92x3 for 2 ≤ x ≤ 14,
x
x
we can show that
|P (Wn ≤ x) − Φ(x)| ≤ 37.408δx .
Subcase 3.2. (1 + x)2 αx < 15 .
Note that for x ≥ 2, we have
0 ≤ αx ≤
1
≤ 0.023 ≤ 0.11.
5(1 + x)2
By (2.7) and Proposition 2.2(2), we obtain the required bounds.
Case 4. x > 14.
Follows the argument of case 3 on replacing the inequalities
x2
e 2 ≥ 60x3 and
in (2.10).
1+x
1
= 1 + ≤ 1.071
x
x
Proof of Corollary 1.4. If 0 ≤ x < 1.3.
We used the same argument as case 1 of Theorem 1.3 and the fact that (1 + x4 )3 ≤ 2.327 to get
C = 9.54.
Suppose that x ≥ 1.3. By the fact that
1 + x4 2
δx ≤
δ x4 ,
1+x
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 92, 10 pp.
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10
K. N EAMMANEE AND P. T HONGTHA
we have

0.332δ x4 if 1.3 ≤ x < 2,





x

 0.250δ 4 if 2 ≤ x < 3,
0.192δ x4 if 3 ≤ x < 7.98,
δx ≤



0.112δ x4 if 7.98 ≤ x < 14,




0.090δ x4 if x ≥ 14.
Then Corollary 1.4 follows from this fact and Theorem 1.3.
R EFERENCES
[1] A. BIKELIS, Estimates of the remainder in a combinatorial central limit theorem, Litovsk. Math.
Sb., 6(3) (1966), 323–346.
[2] A.C. BERRY, The accuracy of the Gaussian approximation to the sum of independent variables,
Trans. Amer. Math. Soc., 49 (1941), 122–136.
[3] L.H.Y. CHEN AND Q.M. SHAO, A non-uniform Berry-Esseen bound via Stein’s method, Probab.
Theory Related Fields, 120 (2001), 236–254.
[4] P.N. CHEN, Asymptotic refinement of the Berry-Esseen constant. Unpublished manuscript (2002)
[5] C.G. ESSEEN, Fourier analysis of distribution functions. A mathematical study of the laplace
Gaussian law, Acta Math., 77 (1945), 1–125.
[6] R. MICHEL, On the constant in the non-uniform version of the Berry-Esseen theorem, Z. Wahrsch.
Verw. Gebiete , 55 (1981), 109-117.
[7] S.V. NAGAEV, Some limit theorems for large deviations, Theory Probab. Appl., 10 (1965), 214–
235.
[8] K. NEAMMANEE, On the constant in the nonuniform version of the Berry-Esseen theorem, International Journal of Mathematics and Mathematical Sciences, 12 (2005), 1951–1967.
[9] L. PADITZ, Ber die Annäherung der Verteilungsfunktionen von Summen unabhängiger Zufallsgröben gegen unberrenzt teilbare Verteilungsfunktionen unter besonderer berchtung der
Verteilungsfunktion de standarddisierten Normalver- teilung. Dissertation, A. TU Dresden,1977.
[10] L. PADITZ, On the analytical structure of the constant in the nonuniform version of the Esseen
inequality, Statistics, 20 (1989), 453–464.
[11] I.S. SIGANOV, Refinement of the upper bound of the constant in the central limit theorem, Journal
of Soviat Mathematics, (1986), 2545–2550.
[12] C. STEIN, A bound for the error in the normal approximation to the distribution of a sum of
dependent random variables, Proc. Sixth Berkeley Symp. Math. Stat. Prob., 2, Univ. California
Press, Berkeley, 1972, 583-602,
[13] C. STEIN, Approximation Computation of Expectation, Hayword California:IMS,1986.
[14] P. THONGTHA AND K. NEAMMANEE, Refinement on the constants in the non-uniform version
of the Berry-Esseen theorem, Thai Journal of Mathematics, 5 (2007), 1–13.
[15] P. VAN BEEK, An application of Fourier methods to the problem of sharpening the Berry-Esseen
inequality, Z. Wahrsch. Verw. Gebiete, 23 (1972), 187–196.
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 92, 10 pp.
http://jipam.vu.edu.au/