International Mathematical Forum, 3, 2008, no. 15, 721 - 738
Berry-Esseen Bound for Independent
Random Sum via Stein’s Method
N. Chaidee and K. Neammanee
Department of Mathematics, Faculty of Science
Chulalongkorn University, Bangkok 10330, Thailand
[email protected]
Abstract
Let (Xn) be a sequence of independent identically distributed random variables with zero means and finite third moments and N a positive integral-valued random variable such that N, X1, X2 , . . . are independent. In this work, we apply a new technique, which is called Stein’s
method, to give uniform and non-uniform bounds in normal approximation of a random sum X1 + X2 + · · · + XN .
Mathematics Subject Classification: 60F05, 60G05
Keywords: normal approximation, Stein’s method, random sum, concentration inequality, uniform and non-uniform bounds
1
Introduction
One of the most important theorem in probability theory and statistics is the
central limit theorem(CLT) which is stated as follows.
Let X1 , X2 , . . . , Xn be independent random variables with EXi = 0 and
EXi2 = σi2 < ∞. Then for
Wn =
X1 + X2 + · · · + Xn
s2n
and s2n = σ12 + σ22 + · · · + σn2
we have
P (Wn ≤ x) → Φ(x)
as n → ∞
where Φ(x) is the standard normal distribution function, that is,
x
1
2
Φ(x) = √
e−t /2 dt.
2π −∞
722
N. Chaidee and K. Neammanee
To apply the CLT we always need to know a bound between P (Wn ≤ x) and
Φ(x), i.e., |P (Wn ≤ x) − Φ(x)|. Berry([2]) and Esseen([6]) are the first two
mathematicians who gave this bound. Their bound is of the form
n
C E|Xi |3 .
sup |P (Wn ≤ x) − Φ(x)| ≤ 3
s
x∈Ê
n i=1
(1)
Many authors([2], [6], [20]) find and improve the constant C in (1) and the best
C is 0.7655 which is obtained by Shiganov in 1986([12]) in case of X1 , X2 , . . . , Xn
are identically and 0.7915 in case of non-identically([12]).
In many situations, we need to approximation the probability of random
sum. Let X1 , X2 , . . . be independent identically distributed random variables
with EXi = 0 and E|Xi |3 = γ < ∞. Let N be a positive integral-valued
random variable with EN < ∞ and N, X1 , X2 , . . . are independent.
Let
W =
X1 + X2 + · · · + XN
√
EN σ
where σ 2 = EXi2 for all i = 1, 2, . . . .
The CLT for W are obtained by many mathematicians(see, [14], [1], [22],
[9], [13], [15], [11] and [5] for examples). In this work, we concerns on uniform
and non-uniform bounds between P (W ≤ x) and Φ(x) by using a new method,
which is called Stein method.
Normally, the method which is used in our literatures is Fourier transform
while in this paper we will use a new method which introduced by Stein in
1972([7]) to obtain bounds of |P (W ≤ x) − Φ(x)|. Stein’s method was introduced by Stein in 1972 for normal approximation of dependent random variables. This method use differential equation which always call Stein’s equation
in stead of Fourier transform. There are three approaches of Stein’s method
for normal approximation, namely, inductive approach(for example, see [4],
[10]), coupling approach(for example, see [16], [8]) and concentration inequality approach(for example, see [21]). The concentration inequality approach
was originally used by Stein for independent and identically distributed random variable([21]) and it was extended by Chen in([18], [19], [17]). In this
paper, we use concentration inequality approach which used in [18] to find
bounds of |P (W ≤ x) − Φ(x)|. A uniform bound in normal approximation is
in section 2 while a non-uniform bound is in section 3.
2
Uniform Bound
Let X1 , X2 , . . . be independent identically distributed random variables with
EXi = 0, EXi2 = σ 2 and E|Xi |3 = γ < ∞. Let N be an integral-valued
723
Berry-Esseen bound for independent random sum
random variable with EN 3/2 < ∞ and N, X1 , X2 , . . . are independent.
Define
Xi
Yi = √
EN σ
and
W = Y1 + Y2 + · · · + YN .
Then EYi = 0 and
2
EW = E(
N
i=1
2
Yi ) =
∞
P (N = n)
n=1
n
EYi2 = 1.
i=1
In this section, we provide a uniform bound between P (W ≤ x) and Φ(x).
Our main result is in Theorem 2.1.
Theorem 2.1. There exists a constant C such that
1.5δEN 3/2 E|N − EN|
0.125
+
+
sup |P (W ≤ x) − Φ(x)| ≤ 10.32δ + √
(EN)3/2
EN
EN
x∈Ê
γ
.
where δ = √
EN σ 3
Corollary 2.2. If N = n, then there exists a constants C such that
sup |P (W ≤ x) − Φ(x)| ≤
x∈
Ê
1
11.82γ
√
√
(1
+
).
σ3 n
n
To prove Theorem 2.1, we need the following concentration inequality.
Proposition 2.3. (Uniform Concentration Inequality). Let
Wn =
n
Yi and Wn(i) = Wn − Yi
for i = 1, 2, . . . , n.
i=1
If δ <
1
, then for a < b
20
P (a ≤
Wn(i)
2.5EN 1
{ (b − a) + δ}
≤ b) ≤
n
2
δ n
+ 2δ + √
.
EN
EN
724
N. Chaidee and K. Neammanee
Proof. In [18], Chen and Shao defined
⎧
1
⎪
⎨− 2 (b − a) − δ
f(t) = t − 12 (b + a)
⎪
⎩1
(b − a) + δ
2
f : R → R by
for t < a − δ,
for a − δ ≤ t ≤ b + δ,
for t ≥ b + δ
and showed that
EWn(i) f(Wn(i) ) ≥ E{I(a ≤ Wn(i) ≤ b)S} − P (a ≤ Wn(i) ≤ b)E(|Yi | min{δ, |Yi|}),
(2)
n
n
EXj2
1 E|Xj |3
−
ES ≥
ENσ 2 2δ j=1 (ENσ 2 )3/2
j=1
(3)
1
|EWn(i) f(Wn(i) )| ≤ { (b − a) + δ}E|Wn(i) |
2
(4)
and
where
S=
n
|Yj | min{δ, |Yj |},
j=1
I(A)(w) =
1 if w ∈ A,
0 if w ∈
/ A.
By (3),
ES ≥
nγ
n
n
−
.
=
2
3/2
EN
2δ(ENσ )
2EN
(5)
From this fact and the fact that
n
2
ES = E(
|Yj | min{δ, |Yj |})2
=
j=1
n
E(|Yj | min{δ, |Yj |})2 +
j=1
≤
n
E(|Yj | min{δ, |Yj |})2 + (
n
j=1
2
=
E[|Yi | min{δ, |Yi|}][|Yj | min{δ, |Yj |}]
i,j
i=j
n
j=1
≤ δ2
j=1
E|Yj |2 + E 2 S
nδ
+ E 2 S,
EN
E|Yj | min{δ, |Yj |})2
Berry-Esseen bound for independent random sum
725
we have
P (S ≤
0.4n
0.4n
) = P (E(S) − S ≥ E(S) −
)
EN
EN
n
≤ P (E(S) − S ≥
)
10EN
100(EN)2
E(S − E(S))2
≤
n2
100(EN)2
=
(ES 2 − E 2S)
n2
5δEN
.
≤
n
Hence, by (2) and the fact that for s < t, y ≥ 0,c > 0,
I(s ≤ w ≤ t)y ≥ c(I(s ≤ w ≤ t) − (1 − y/c)I(y ≤ c)),
(6)
we have
0.4n
2.5SEN
0.4n
{P (a ≤ Wn(i) ≤ b) − E[(1 −
)I(S ≤
)]}
EN
n
EN
− P (a ≤ Wn(i) ≤ b)E(|Yi | min{δ, |Yi |})
0.4n
0.4n
≥
{P (a ≤ Wn(i) ≤ b) − EI(S ≤
)} − δE|Yi|
EN
EN
δ
0.4n
0.4n
{P (a ≤ Wn(i) ≤ b) − P (S ≤
)} − √
≥
EN
EN
EN
0.4n
δ
P (a ≤ Wn(i) ≤ b) − 2δ − √
≥
.
EN
EN
EWn(i) f(Wn(i) ) ≥
This implies,
2.5EN δ EWn(i) f(Wn(i) ) + 2δ + √
n
EN
δ 2.5EN 1
(i)
( (b − a) + δ) E|Wn |2 + 2δ + √
≤
n
2
EN
2.5EN 1
δ n
( (b − a) + δ)
≤
+ 2δ + √
n
2
EN
EN
P (a ≤ Wn(i) ≤ b) ≤
where we have used (4) in the second inequality.
Proof of Theorem 2.1.
To bound P (W ≤ x) − Φ(x), it suffices to consider x ≥ 0 as we can apply
result to −W when x < 0. By [18], p. 246, we have
|P (W ≤ x) − Φ(x)| ≤ 0.55 for x ≥ 0,
726
N. Chaidee and K. Neammanee
hence we only need to consider the case where
δ<
1
.
20
Let f be the unique bounded solution fx of the Stein equation
f (w) − wf (w) = I(w ≤ x) − Φ(x).
(7)
If we replace w in (7) by Wn , then
Ef (Wn ) − EWn f(Wn ) = P (Wn ≤ x) − Φ(x).
Set pn = P (N = n). Then, by the fact that
∞
n
EWn f(Wn ) =
E
f (Wn(i) + t)Mi(t)dt
−∞
i=1
(eq. (2.17), p.12 of [3]) where
Mi (t) = E[Yi (I(0 ≤ t ≤ Yi ) − I(Yi ≤ t < 0))],
yields
P (W ≤ x) − Φ(x)
∞
∞
pn P (Wn ≤ x) −
pn Φ(x)
=
n=1
=
=
∞
n=1
∞
n=1
pn [P (Wn ≤ x) − Φ(x)]
pn [Ef (Wn ) − EWn f(Wn )]
n=1
=
∞
pn
n=1
∞
+
n
∞
E
−∞
i=1
f (Wn )Mi (t)dt −
pn Ef (Wn ) −
n
n=1
−∞
∞
E
−∞
i=1
∞
E
i=1
n
f (Wn(i) + t)Mi (t)dt
f (Wn )Mi (t)dt
= A1 + A2
(8)
where
A1 =
∞
pn
n=1
n
i=1
∞
E
−∞
f (Wn )Mi (t)dt −
∞
n
pn Ef (Wn ) −
E
A2 = n=1
i=1
n
i=1
∞
−∞
∞
E
−∞
f (Wn(i) + t)Mi (t)dt ,
f (Wn )Mi (t)dt .
727
Berry-Esseen bound for independent random sum
By noting that
⎧
⎪
if w + s ≤ x, w + t > x,
⎨1
f (w + s) − f (w + t) ≤ (|w| + 0.63)(|s| + |t|) if s ≥ t,
⎪
⎩
0
otherwise
(see [18], p. 247), we have
A1 =
≤
∞
n=1
∞
n
pn
E
n
n=1
≤
≤
∞
pn
n
n=1
i=1
+
pn
∞
E
(i)
Wn +t>x
(i)
Wn +Yi ≤x
E
E
Yi ≥t
n=1
i=1
n
∞
EE Yi
+
∞
n=1
∞
pn
pn
n
i=1
n
n=1
i=1
n=1
i=1
∞
E
i=1
E
Yi ≥t
(|Wn(i) | + 0.63)(|Y i| + |t|)Mi(t)dt
(|Wn(i) | + 0.63)(|Y i| + |t|)Mi(t)dt
t>Yi
i=1
Mi (t)dt +
n
I(x − t < Wn(i) ≤ x − Yi )Mi (t)dt
t>Yi
n
pn
f (Wn(i) + Yi ) − f (Wn(i) + t) Mi (t)dt
i=1
n=1
≤
−∞
i=1
pn
∞
I(x − t < Wn(i) ≤ x − Yi )Mi (t)dt
E(|Wn(i) | +
∞
0.63)E
−∞
(|Yi | + |t|)Mi (t)dt
E Yi I(x − t < Wn(i) ≤ x − Yi )Mi (t)dt
t>Yi
n 1
(i)
pn
( E(Wn )2 + 0.63)(E |Yi|E|Yi|2 + E|Yi |3 )
+
2
n=1
i=1
∞
n
pn
E
P (x − t < Wn(i) ≤ x − Yi |Yi )Mi (t)dt
≤
∞
3
+
2
≤
∞
n=1
t>Yi
pn
n=1
n
pn
i=1
n
(
i=1
E
∞
−∞
n
+ 0.63)E |Yi|3
EN
P (x − t < Wn(i) ≤ x − Yi |Yi )Mi (t)dt
∞
3γ
pn n(
+
2(EN )3/2σ 3 n=1
n
+ 0.63)
EN
728
N. Chaidee and K. Neammanee
2.5EN 1
δ n
≤
Mi (t)dt
{ (|t| + |Yi |) + δ}
+ 2δ + √
pn
E
n
2
EN
EN
t>Yi
n=1
i=1
∞
n
0.945γEN
1.5γEN 3/2
+
2
3
(EN ) σ
(EN )3/2σ 3
n ∞
2.5EN 1 E|Yi|3
n
2
2
≤
pn
{ (
+ E|Yi|E|Yi| ) + δE|Yi| }
+ 2δE|Yi|2
n
2
2
EN
n=1
i=1
δE|Yi|2 1.5δEN 3/2
+ √
+ 0.945δ
+
(EN )3/2
EN
+
2.5δ
1.5δEN 3/2
= 9.375δ + √
+
+ 0.945δ
(EN )3/2
EN
1.5δEN 3/2
0.125
+
≤ 10.32δ + √
(EN )3/2
EN
(9)
where we have used Proposition 2.3 in the seventh
Similar to (9), we can use the fact that
⎧
⎪
⎨−1
f (w + s) − f (w + t) ≥ −(|w| + 0.63)(|s| + |t|)
⎪
⎩
0
inequality.
if w + s > x, w + t ≤ x,
if s < t,
otherwise,
(see [18], p. 247) to prove that
1.5δEN 3/2
0.125
A1 ≥ −10.32δ − √
−
.
(EN)3/2
EN
Hence
1.5δEN 3/2
0.125
|A1| ≤ 10.32δ + √
+
.
(EN)3/2
EN
By [18], p. 247,
|f (w)| ≤ 1 for all w,
(10)
then
∞
n
|A2| = pn Ef (Wn ) 1 −
E
≤
n=1
∞
n
n=1
i=1
pn 1 −
EYi2 n =
pn 1 −
EN
n=1
∞
i=1
∞
−∞
Mi (t)dt Berry-Esseen bound for independent random sum
N = E 1 −
EN
E|N − EN|
=
.
EN
By combining (8)-(11), we have
1.5δEN 3/2 E|N − EN|
0.125
√
+
+
|P (W ≤ x) − Φ(x)| ≤ 10.32δ +
.
(EN)3/2
EN
EN
3
729
(11)
2
Non-uniform Bound
In this section, we will give a non-uniform bound in normal approximation of
W = Y1 + Y2 + · · · + YN
and further assume that EN 5 < ∞.
Theorem 3.1. There exists a constant C such that for every real number x,
EN 5 C (V arN )EN 2
Cδ 1+
+
.
|P (W ≤ x) − Φ(x)| ≤
(1 + |x|)3
(EN)5
(1 + |x|)2(EN)2
Corollary 3.2. If N = n, then there exists a constants C such that for
every real number x,
Cδ
|P (W ≤ x) − Φ(x)| ≤
.
(1 + |x|)3
The following proposition concerns a concentration inequality for a nonuniform bound.
Proposition 3.3. (Non-uniform Concentration Inequality) For 0 ≤ a < b,
we have
n4 C(b − a + δ) EN
+
.
P (a ≤ Wn(i) ≤ b) ≤
(1 + a)3
n
(EN)4
where C is a constant.
Proof. In the proof of non-uniform concentration inequality in [18], p. 241,
Chen and Shao defined
Zj = Yj I(|Yj | ≤ 1 + a) − EYj I(|Yj | ≤ 1 + a),
n
Zj and
Tn(i) =
r=
j=1
j=i
n
j=1
j=i
EYj I(|Yj | > 1 + a),
730
N. Chaidee and K. Neammanee
and showed that
P (a ≤ Wn(i) ≤ b) ≤ P (a + r ≤ Tn(i) ≤ b + r) + P ( max |Yj | > 1 + a).
1≤j≤n
Hence, by the fact that
n
1
nδ
P ( max |Yj | > 1 + a) ≤
E|Yj |3 =
,
3
1≤j≤n
(1 + a) j=1
EN(1 + a)3
we have
P (a ≤ Wn(i) ≤ b) ≤ P (a + r ≤ Tn(i) ≤ b + r) +
nδ
.
EN(1 + a)3
(12)
To prove this proposition, it suffices to bound
P (a + r ≤ Tn(i) ≤ b + r).
By Rosenthal inequality(see, for example, [23], p.59) and the fact that
n
E|Tn(i) |2 ≤
,
EN
we have
n
(i) 4
(i) 2 2
E|Yj |4I(|Yj | ≤ 1 + a)
E|Tn | ≤ C (E|Tn | ) +
(13)
j=1
j=i
2 2
3
≤ C (nEY1 ) + n(1 + a)E|Y1 |
n2
n(1 + a)δ +
.
≤C
(EN)2
EN
(14)
Case 1. (1 + a)δ ≥ 1.
By the fact that r ≤
n
1
and
< δ, we have
EN
1+a
P (a + r ≤ Tn(i) ≤ b + r) ≤ P (1 + a ≤ 1 − r + Tn(i) )
(i)
≤
E|1 − r + Tn |4
(1 + a)4
(i)
≤
≤
≤
≤
≤
C(1 + r 4 + E|Tn |4 )
(1 + a)4
n2
C
Cn4
C
n(1 + a)δ +
+
+
(1 + a)4 (EN )4(1 + a)4 (1 + a)4 (EN )2
EN
2
4
n
n
n
Cδ
+
+
1+
3
2
(1 + a)
EN
(EN )
(EN )4
n4
Cδ
{1 +
}
3
(1 + a)
(EN )4
n4
EN
Cδ
+
{
}.
(15)
(1 + a)3 n
(EN )4
731
Berry-Esseen bound for independent random sum
Case 2. (1 + a)δ < 1.
n−1
Follows the argument in [18], p. 243 and choose C in (6) is
, we can
4EN
show that
E(Tn(i) f(Tn(i) )) ≥
where U =
n
(n − 1)(1 + a)3 n−1 P (a + r ≤ Tn(i) ≤ b + r) − P (U ≤
)
4EN
4EN
(16)
ηj and ηj = |Zj | min(δ, |Zj |) and
j=1
j=i
⎧
⎪
for x < a + r − δ,
⎨0
f(x) = (1 + x − r + δ)3(x − a − r + δ) for a + r − δ ≤ x ≤ b + r + δ,
⎪
⎩
for x > b + r + δ.
(1 + x − r + δ)3(b − a + 2δ)
d2
and EYi = 0, we have
By the fact that for any c, d > 0, min(c, d) ≥ d −
4c
EU ≥
=
n
|Zj |3 E Zj2 −
4δ
j=1
j=i
n EYj2I(|Yj | ≤ 1 + a) − (EYj I(|Yj | ≤ 1 + a))2
j=1
j=i
732
N. Chaidee and K. Neammanee
−
=
n
3
1 E Yj I(|Yj | ≤ 1 + a) − EYj I(|Yj | ≤ 1 + a)
4δ j=1
j=i
n EYj2I(|Yj | ≤ 1 + a) − (EYj I(|Yj | > 1 + a))2
j=1
j=i
n
3
1 E Yj I(|Yj | ≤ 1 + a) − EYj I(|Yj | ≤ 1 + a)
−
4δ j=1
≥
n j=i
EYj2 I(|Yj |
≤ 1 + a) −
EYj2 I(|Yj |
> 1 + a)
j=1
j=i
1 3
3
E |Yj | I(|Yj | ≤ 1 + a) + |EYj I(|Yj | ≤ 1 + a)|
−
δ j=1
n
j=i
≥
n
j=1
j=i
EYj2
2
−
E|Yj |3
δ j=1
n
j=i
2
(n − 1)σ
2(n − 1)γ
−
ENσ 2
δ(ENσ 2)3/2
n−1
=
.
2EN
=
From this fact and Rosenthal inequality,
P (U ≤
n−1 n−1
n−1
) ≤ P (EU − U ≥
−
)
4EN
2EN
4EN
n−1
)
= P (EU − U ≥
4EN
16(EN)2
≤
E|U − E(U)|4
(n − 1)2
n
n
C(EN)2 2 2 4 ≤
Eηj +
Eηj
(n − 1)2
j=1
j=1
2 j=i
j=i
2
C(EN)
3
4
4
(n − 1)δE|Z1 |
+ (n − 1)δ E|Z1 |
≤
(n − 1)2
2
C(EN)2 3
4
3
+ (n − 1)(1 + a)δ E|Y1 |
(n − 1)δE|Y1 |
≤
(n − 1)2
Berry-Esseen bound for independent random sum
C(EN)2 (n − 1)δ 2 2 (n − 1)(1 + a)δ 5 +
(n − 1)2
EN
EN
5
(1 + a)δ EN
= C δ4 +
n−1
Cδ EN ≤
1
+
(1 + a)3
n−1
Cδ
EN ≤
1
+
(1 + a)3
n
733
=
(17)
where we have used the fact that (1 + a)δ < 1 in the last inequality.
By (13), (14), (16) and (17),
P (a + r ≤ Tn(i) ≤ b + r)
4EN
n−1
)+
≤ P (U ≤
E(Tn(i) f(Tn(i) ))
3
4EN
(n − 1)(1 + a)
EN Cδ CEN
{(b − a + 2δ)E|Tn(i) (1 + Tn(i) − r + δ)3|}
1
+
≤
+
3
3
(1 + a)
n
n(1 + a)
CEN
EN Cδ +
1
+
≤
{(b − a + 2δ)((1 + r3 )E|Tn(i) | + E|Tn(i) |4)}
(1 + a)3
n
n(1 + a)3
EN Cδ 1
+
≤
(1 + a)3
n
CEN
n3
n2
n(1 + a)δ (i) 2
+
(b
−
a
+
δ){1
+
E|T
|
+
+
}
n
n(1 + a)3
(EN)3
(EN)2
EN
EN Cδ 1
+
≤
(1 + a)3
n
√
CEN
n
n7/2
n2
n
+
+
(b − a + 2δ){ √
+
}
+
n(1 + a)3
EN
(EN)2 (EN)7/2
EN
EN CEN(b − a + δ) n4 Cδ +
1
+
1
+
≤
(1 + a)3
n
n(1 + a)3
(EN)4
n3 C(b − a + δ) EN
+
.
(18)
≤
(1 + a)3
n
(EN)3
By (12), (15) and (18), we have
P (a ≤
Wn(i)
C(b − a + δ) EN
n4 ≤ b) ≤
.
+
(1 + a)3
n
(EN)4
734
N. Chaidee and K. Neammanee
Lemma 3.4. Let fx be the solution of (7) and g(w) = (wfx (w)) .
1. For x > 0, there exists a constant C such that
E|fx (Wn )| ≤
n C .
1
+
(1 + x)2
EN
x
2. For x ≥ 4 and |u| ≤ 1 + , we have
4
EN
C n4 (1
+
δx)
+
Eg(Wn(i) + u) ≤
(1 + x)3
n
(EN)4
for i = 1, 2, . . . , n.
Proof.
1. Using the argument of Lemma 5.1 in [18], p.248 and the fact that
EWn2 =
n
,
EN
we have
E|fx (Wn )| ≤
C n .
1
+
(1 + x)2
EN
2. Use the argument of Lemma 5.2 in [18], p.248 and Proposition 3.3.
Proof of Theorem 3.1.
In view of Theorem 2.1, we may without loss of generality, assume x ≥ 4.
If (1 + x)δ ≥ 1, then, the arguments in (14) and (15) yield
P (W ≥ x) ≤
EN 4 Cδ 1
+
(1 + x)3
(EN)4
which implies
|F (x) − Φ(x)| ≤ P (W ≥ x) + 1 − Φ(x)
EN 4 C
Cδ 1
+
+
≤
3
4
(1 + x)
(EN)
(1 + x)4
EN 4 Cδ 1
+
.
≤
(1 + x)3
(EN)4
Hence, we assume (1 + x)δ < 1.
Berry-Esseen bound for independent random sum
735
By the same argument as (8), we have
F (x) − Φ(x)
∞
n
∞
x
pn
E I(|Yi | ≤ 1 + )
(f (Wn(i) + Yi ) − f (Wn(i) + t))Mi (t)dt
=
4 −∞
n=1
i=1
∞
n
∞
x
pn
E I(|Yi| > 1 + )
(f (Wn(i) + Yi ) − f (Wn(i) + t))Mi(t)dt
+
4 −∞
n=1
i=1
∞
∞
n
pn Ef (Wn ) −
E
f (Wn )Mi (t)dt
+
n=1
i=1
−∞
:= A1 + A2 + A3.
One can follows the argument of [18] on p.251-252 to show that
Yi
∞
n ∞
x
pn
Mi (t)
Eg(Wn(i) + u)dudt |A1| ≤
E I(|Yi| ≤ 1 + )
4 −∞
t
n=1
i=1
n
∞
x
pn
E I(|Yi| ≤ 1 + )
+
4
n=1
i=1
∞
×
P (x − max(Yi , t) < Wn(t) ≤ x − min(Yi , t)|Yi)Mi (t)dt
−∞
= A11 + A12 .
Then Proposition 3.3 and Lemma 3.4(2) yields
|A11 | ≤
≤
≤
=
=
=
n ∞
C
x
pn
E I(|Yi | ≤ 1 + )
3
(1 + x) n=1 i=1
4
∞
Yi
EN
n4 ×
dudt
Mi (t)
(1 + δx)
+
4
n
(EN)
−∞
t
n
∞
n4 C(1 + δx) EN
+
pn
E|Yi |3
(1 + x)3 n=1
n
(EN)4 i=1
∞
EN
n4 nδ
C
+
p
n
(1 + x)3 n=1
n
(EN)4 EN
∞
n5 Cδ
EN
+
p
n
(1 + x)3EN n=1
(EN)4
EN 5 Cδ
EN +
(1 + x)3EN
(EN)4
EN 5 Cδ
1+
(1 + x)3
(EN)5
736
N. Chaidee and K. Neammanee
where we have used the fact that (1 + x)δ < 1 in the third inequality, and
∞
EN
n4 C
+
|A12| ≤
pn
(1 + x)3 n=1
n
(EN)4
n
x
×
E I(|Yi | ≤ 1 + )
(|t| + |Yi | + δ)Ki (t)dt
4 |t|≤1+ x4
i=1
n
∞
EN
C
n4 pn
(E|Yi |3 + δE|Yi |2)
≤
+
(1 + x)3 n=1
n
(EN)4 i=1
∞
EN
C
n4 nδ
nδ
(
pn
=
+
+
)
3
4
(1 + x) n=1
n
(EN)
EN
EN
∞
n5 Cδ
pn EN +
=
(1 + x)3 EN n=1
(EN)4
Cδ EN 5 =
1+
.
(1 + x)3
(EN)5
Hence
|A1| ≤
EN 5 Cδ 1
+
.
(1 + x)3
(EN)5
By (10), we have
|A2| ≤
∞
pn
n=1
∞
≤C
n=1
n
x
P (|Yi | > 1 + )
4
i=1
pn
n
E|Yi |3
(1 + x)3
i=1
∞
C
nδ
≤
pn
3
(1 + x) n=1 EN
≤
Cδ
.
(1 + x)3
Follows the argument of (11) and Lemma 3.4(1), we have
n
n C
n |A3| ≤
1
+
p
−
1
n
(1 + x)2 i=1
EN
EN
CE|N 2 − (EN)2 |
(1 + x)2 (EN)2
C (V arN )EN 2
≤
.
(1 + x)2 (EN)2
=
Berry-Esseen bound for independent random sum
737
Hence,
EN 5 C (V arN )EN 2
Cδ 1+
+
.
|P (W ≤ x) − Φ(x)| ≤
(1 + x)3
(EN)5
(1 + x)2(EN)2
2
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