Journal of Inequalities in Pure and
Applied Mathematics
AN INEQUALITY FOR THE ASYMMETRY OF DISTRIBUTIONS AND
A BERRY-ESSEEN THEOREM FOR RANDOM SUMMATION
volume 7, issue 1, article 2,
2006.
HENDRIK KLÄVER AND NORBERT SCHMITZ
Institute of Mathematical Statistics
University of Münster
Einsteinstr. 62
D-48149 Münster, Germany.
Received 16 March, 2004;
accepted 15 December, 2005.
Communicated by: S.S. Dragomir
EMail: schmnor@math.uni-muenster.de
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
056-04
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Abstract
We consider random numbers Nn of independent,
identically distributed (i.i.d.)
PNn
random variables Xi and their sums i=1 Xi . Whereas Blum, Hanson and
Rosenblatt [3] proved a central limit theorem for such sums and Landers and
Rogge [8] derived the corresponding approximation order, a Berry-Esseen type
result seems to be missing. Using an inequality for the asymmetry of distributions, which seems to be of its own interest, we prove, under the assumption
E|Xi |2+δ < ∞ for some δ ∈ (0, 1] and Nn /n → τ (in an appropriate sense), a
Berry-Esseen theorem for random summation.
2000 Mathematics Subject Classification: 60E15, 60F05, 60G40.
Key words: Random number of i.i.d. random variables, Central limit theorem for random sums, Asymmetry of distributions, Berry-Esseen theorem for random sums.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
An Inequality for the Asymmetry of Distributions . . . . . . . . . 6
3
Some Futher Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4
A Berry-Esseen Theorem for Random Sums . . . . . . . . . . . . . . 18
References
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
Hendrik Kläver and Norbert
Schmitz
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Page 2 of 25
J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
1.
Introduction
One of the milestones of probability theory is the famous theorem of BerryEsseen which gives uniform upper bounds for the deviation from the normal
distribution in the Central Limit Theorem:
Let {Xn , n ≥ 1} be independent random variables such that
n
X
2
2
2
EXn = 0, EXn =: σn ,
sn :=
σi2 > 0,
Γ2+δ
:=
n
n
X
i=1
E|Xi |2+δ < ∞
i=1
Pn
for some δ ∈ (0, 1] and Sn = i=1 Xi , n ≥ 1. Then there exists a universal
constant Cδ such that
2+δ
Γn
S
n
,
sup P
≤ x − Φ(x) ≤ Cδ
sn
sn
x∈R
where Φ denotes the cumulative distribution function of a N (0, 1)-(normal)
distribution (see e.g. Chow and Teicher [5, p. 299]).
For the special case of identical distributions this leads to:
Let {Xn , n ≥ 1} be i.i.d. random variables with EXn = 0, EXn2 =: σ 2 >
0, E|Xn |2+δ =: γ 2+δ < ∞ for some δ ∈ (0, 1]. Then there exists a universal
constant cδ such that
S
cδ γ 2+δ
n
√ ≤ x − Φ(x) ≤ δ/2
sup P
.
n
σ
σ n
x∈R
Van Beek [2] showed that C1 ≤ 0.7975; bounds for other values Cδ are given
by Tysiak [12], e.g. C0.8 ≤ 0.812; C0.6 ≤ 0.863, C0.4 ≤ 0.950, C0.2 ≤ 1.076.
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
Hendrik Kläver and Norbert
Schmitz
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Page 3 of 25
J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
On the other hand, there exist also central limit theorems for random summation, e.g. the theorem of Blum, Hanson and Rosenblatt [3] which generalizes
previous results by Anscombe [1] and Renyi [11]:
Let {XP
n , n ≥ 1} be i.i.d. random variables with EXn = 0, Var Xn =
1, Sn := ni=1 Xi and let {Nn , n ≥ 1} be N-valued random variables such
P
that Nn /n −→ U where U is a positive random variable. Then
P S Nn /
√
Nn
d
−→ N (0, 1).
Therefore, the obvious question arises whether one can prove also BerryEsseen type inequalities for random sums. A first result concerning the approximation order is due to Landers and Rogge [8] for random variables Xn with
E|Xn |3 < ∞; this was generalized by Callaert and Janssen [4] to the case that
E|Xn |2+δ < ∞ for some δ ∈ (0, 1]:
Let {Xn , n ≥ 1} be i.i.d. random variables with E Xn = µ, Var Xn =
2
σ > 0, and E|Xn |2+δ < ∞ for some δ > 0. Let {Nn , n ≥ 1} be N-valued
random variables, {εn , n ≥ 1} positive real numbers with εn −→ 0 where, for
n→∞
n large, n
such that
then
and
−δ
−1
≤ εn if δ ∈ (0, 1] and n ≤ εn if δ ≥ 1. If there exists a τ > 0
Nn
√
P − 1 > εn = O( εn ),
nτ
!
PNn
√
(X
−
µ)
i
i=1
√
sup P
≤ x − Φ(x) = O( εn )
σ nτ
x∈R !
PNn
√
(X
−
µ)
i
i=1
√
sup P
≤ x − Φ(x) = O( εn ).
σ Nn
x∈R An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
Hendrik Kläver and Norbert
Schmitz
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
Moreover, there exist several further results on convergence rates for random
sums (see e.g. [7] and the papers cited there); as applications, sequential analysis, random walk problems, Monte Carlo methods and Markov chains are mentioned.
However, rates of convergence without any knowledge about the factors are
of very limited importance for applications. Hence the aim of this paper is to
prove a Berry-Esseen type result for random sums i.e. a uniform approximation
with explicit constants. Obviously, due to the dependencies on the moments of
the Xn as well as on the asymptotic behaviour of the sequence Nn such a result
will necessarily be more complex than the original Berry-Esseen theorem.
For the underlying random variables Xn we make the same assumption as in
the special version of the Berry-Esseen theorem:
(
Xn , n ≥ 1, are i.i.d. random variables with EXn = 0,
(M) :
V ar Xn = 1 and γ 2+δ := E|Xn |2+δ < ∞ for some δ ∈ (0, 1].
Similarly as Landers and Rogge [8] or Callaert and Janssen [4] resp. we assume
on the random indices


 Nn , n ≥ 1, are integer-valued random variables and ζn , n ≥ 1,
real numbers with limn→∞ ζn = 0 such that there exist
(R) :

√

d, τ > 0 with P (| Nnτn − 1| > ζn ) ≤ d ζn .
As they are essential for applications, e.g. in sequential analysis, arbitrary dependencies between the indices and the summands are allowed.
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
Hendrik Kläver and Norbert
Schmitz
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
2.
An Inequality for the Asymmetry of
Distributions
A main tool for deriving explicit constants for the rate of convergence is an
inequality which seems to be of its own interest. For a smooth formulation
we use (for the different values of the moment parameter δ ∈ (0, 1]) some
(technical) notation: For ϑ := 2/(1 + δ) and y ≥ 1 let gδ (y) be defined by

p
2

2y
−
1
+
2y
y2 − 1
if δ = 1




n

p

ϑ+1


min
2ϑ y 4ϑ − 1 + 2 2 y 2ϑ 2ϑ−1 y 4ϑ − 1 ,


o

p


8ϑ
4ϑ
8ϑ

2y − 1 + 2y
y −1
if δ ∈ 13 , 1
gδ (y) :=
n
p
ϑ+1

ϑ (2+2k )ϑ
(1+2k−1 )ϑ

2 y

min
2
y
−
1
+
2
2ϑ−1 y (2+2k )ϑ − 1,


o

p

k+1 )ϑ

(4+2k+1 )ϑ
(2+2k )ϑ
(4+2

2y
− 1 + 2y
y
−1


i



1

if δ ∈
, 1
, k≥2
2k +1
2k−1 +1
Theorem 2.1. Let X be a random variable with EX = 0, Var X = 1 and
γ 2+δ := E|X|2+δ < ∞ for some δ ∈ (0, 1]. Then
P (X < 0) ≤ gδ (γ
2+δ
)P (X > 0) and P (X > 0) ≤ gδ (γ
2+δ
)P (X < 0).
Proof. Let X be a random variable as described above. Let X+ = max{X, 0},
X− = min{X, 0}, E(X+ ) = E(X− ) = α and P (X > 0) = p, P (X = 0) =
r, P (X < 0) = 1 − r − p. Since E(X) = 0, Var(X) = 1 it is obvious that
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
Hendrik Kläver and Norbert
Schmitz
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
p, 1 − r − p > 0. As α = pE(|X| |X > 0), we have E(|X| |X > 0) = αp ;
α
.
analogously E(|X| |X < 0) = 1−r−p
Applying Jensen’s inequality to the convex function f : [0, ∞) → [0, ∞),
3+δ
f (x) = x 2 yields
3+δ
3+δ
2
3+δ
3+δ
α 2
α
E(|X| 2 |X > 0) ≥
and E(|X| 2 |X < 0) ≥
.
p
1−r−p
Defining βz := E|X|z for z > 0 we get
3+δ
3+δ
β 3+δ = pE |X| 2 |X > 0 + (1 − r − p)E |X| 2 |X < 0
2
≥α
3+δ
2
(1 − r − p)
1+δ
2
+p
(p(1 − r − p))
1+δ
2
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
Hendrik Kläver and Norbert
Schmitz
1+δ
2
and, therefore,
(i)
α
3+δ
2
≤ β 3+δ
2
(p(1 − r − p))
(1 − r − p)
1+δ
2
1+δ
2
+p
1+δ
2
Since γ 2+δ < ∞, we can apply the Cauchy-Schwarz-inequality to |X|1/2 and
|X|1+δ/2 and obtain
2
3+δ
2
≤ E|X| E|X|2+δ = 2αγ 2+δ ;
E|X|
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hence
(ii)
α≥
β 3+δ
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2
2
2γ 2+δ
.
Page 7 of 25
J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
Combining (i) and (ii) we obtain

2  3+δ
2
1+δ
β
3+δ
(p(1 − r − p)) 2


2
≤ β 3+δ 1+δ


1+δ ;
2
2γ 2+δ
p 2 + (1 − r − p) 2
hence
1
1
x2 ≤ −
4 1−r
ϑ+1
1
a1
2
a2
1
+x
2
ϑ1
+
1
−x
2
ϑ1 !ϑ
with
ϑ+2
2
,
ϑ=
, a1 = β 3+δ
2
1+δ
Obviously x ∈ − 12 , 12 , ϑ ∈ [1, 2).
Since
x=
(iii)
p
1
− ,
1−r 2
a2 = (γ 2+δ )ϑ+1 .
xµ + y µ ≥ (x + y)µ ∀µ ∈ (0, 1], x, y ≥ 0
and 0 < 1 − r ≤ 1 it follows altogether that
s
ϑ−1
1
1
a1
(iv)
|x| ≤
1−
.
2
2
a2
For large values of |x| this estimation is rather poor, so we notice, furthermore,
that
ϑ1 ϑ1 !ϑ
1
1
+x
+
−x
≥ 2ϑ−1 (1 − 4x2 )1/2 ;
2
2
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
Hendrik Kläver and Norbert
Schmitz
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
hence
1
|x| ≤
2
(v)
s
a1
a2
2
s
a2
a1
1−
.
From (iv) and (v) it follows that
(vi)
ϑ+1
a2
p
≤ 2ϑ − 1 + 2 2
1−r−p
a1
a2
2ϑ−1
a1
−1
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
and
(vii)
p
≤2
1−r−p
a2
a1
2
a2
−1+2
a1
s
a2
a1
2
− 1.
Hendrik Kläver and Norbert
Schmitz
4
3+δ
≤ σ2 =
1. Let δ = 1. Then β 3+δ = σ 2 = 1 and so aa12 = (γ 3 )2 . Let δ ∈ 3 , 1 . Then
2
5−δ
≤
2
+
δ,
so
β
5−δ exists. Due to the Cauchy-Schwarz-inequality we have
2
Now we estimate β 3+δ . Due to Jensen’s inequality we have β 3+δ
2
1
2
1 = (σ 2 )2 ≤ β 3+δ β 5−δ ;
2
2
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hence
β 3+δ ≥
2
1
β 5−δ
2
altogether
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2
≥
1
5−δ
(γ 2+δ ) 2(2+δ)
a2
≤ (γ 2+δ )4ϑ .
a1
.
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
Let k ∈ N, k ≥ 2. For δ ≥ 2k1+1 it follows that 2 + 1−δ
≤ 2 + δ; hence β2+ 1−δ
2k
2k
exists. Due to the Cauchy-Schwarz-inequality we have
1 = (σ 2 )2 ≤ β2+ 1−δ β2− 1−δ
2k
and
β2− 1−δ
j+1
2j
=
2
β2− 1−δ
j+1
2
for j ∈ {1, . . . , k − 1}. This yields
2k−1
≥
β 3+δ ≥ β2− 1−δ
2k
2
2 2j−1
2k
2j−1
≤ β2− 1−δ
β
2
j
2
1
β2+ 1−δ
2k−1 ≥
1
(γ 2+δ )
2k+1 +1−δ
2(2+δ)
.
Hendrik Kläver and Norbert
Schmitz
2k
h
k
Altogether we get aa12 ≤ (γ 2+δ )(2 +2)ϑ for δ ∈ 2k1+1 , 2k−11 +1 .
Combining this with (vi) and (vii) we obtain the assertion.
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Remark 1. For each δ ∈ (0, 1] equality holds in Theorem 2.1 iff P X = 12 (ε1 +
ε−1 ) (where εx denotes the Dirac measure in x).
Proof. (i) Let X be a real random variable with P X = 12 (ε1 + ε−1 ). Then
E(X) = 0, Var(X) = 1, γ 2+δ = 1 and so gδ (γ 2+δ ) = 1 for all δ ∈ (0, 1].
Since P (X < 0) = P (X > 0) = 21 we get equality.
2
2γ 2+δ
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(ii) In Theorem 2.1 we have
β 23+δ
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
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! 3+δ
2
≤α
3+δ
2
≤
(p(1 − r − p))
(1 − r − p)
1+δ
2
1+δ
2
+p
1+δ
2
β 3+δ .
Page 10 of 25
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
In the first “≤” there is equality iff |X|1/2 and |X|1+δ/2 are linearly dependent
P -almost surely, i.e. P (|X| ∈ {0, c}) = 1 for some c > 0. As E(X) = 0, we
obtain P X = p(εc +ε−c )+(1−2p)ε0 . So the inequality is sharp iff gδ (γ 2+δ ) = 1.
With 1 = E(X 2 ) = 2c2 p we obtain
γ 2+δ = 2pc2+δ =
1
.
(2p)δ/2
The functions hδ : (0, 12 ] → R, δ ∈ (0, 1], defined by hδ (p) = gδ (2p)1δ/2 , are
strictly decreasing. Due to p ∈ 0, 21 and hδ 12 = 1 we get p = 12 , r = 0 and
c = 1, therefore P X = 21 (ε1 + ε−1 ).
Since the Central Limit Theorem is concerned with sums of random variables (instead of single variables) we need a corresponding generalization of
Theorem 2.1.
Corollary 2.2. Under assumption (M),

2+δ 
n
1 X
γ 2+δ − 1
E  √
Xi  ≤
+ n2−δ/2
δ/2
n
n
i=1
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
Hendrik Kläver and Norbert
Schmitz
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holds, and therefore,
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P
n
X
i=1
!
Xi < 0
≤ gδ
γ
2+δ
−1
nδ/2
+ n2−δ/2 P
n
X
i=1
!
Xi > 0 .
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Page 11 of 25
J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
Proof.

n
3  2+δ
3
n
X 2+δ
X
Xi = 
Xi 
i=1
i=1
 2+δ
3

n
n
n
X
X
X

3
2
≤
|Xi | + 3
|Xk Xi | +
|Xi Xk Xl |
i=1
≤
n
X
i=1
|Xi |2+δ + 3
k,i=1
k6=i
n
X
k,i=1
k6=i
i,k,l=1
i6=k6=l6=i
|Xk2 Xi |
2+δ
3
+
n
X
|Xi Xk Xl |
2+δ
3
.
i,k,l=1
i6=k6=l6=i
Since the Xi are independent and, due to E|Xi |2 = 1 and Jensen’s inequality,
E|Xi |α ≤ 1 for α ≤ 2, we obtain
2+δ
n
1 X
γ 2+δ − 1
1
+ n2−δ/2 .
E √
Xi ≤ 1+δ/2 (nγ 2+δ + (n3 − n)) =
δ/2
n
n
n
i=1
P
2
P
= 1 and Theorem 2.1 the
From E √1n ni=1 Xi = 0, E √1n ni=1 Xi
assertion follows.
We need a bound which does not depend on the number of summands. For
this aim we use for n < κ := (4cδ γ 2+δ )2/δ the asymmetry inequality of Corollary 2.2 and for n ≥ κ the Berry-Esseen bound (the choice of κ as boundary
between “small” and “large” n is somewhat arbitrary).
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
Hendrik Kläver and Norbert
Schmitz
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
Corollary 2.3. Under Assumption (M)
!
n
X
P
Xi < 0 ≤ (fδ (γ 2+δ ) − 1)P
i=1
n
X
!
Xi > 0 ,
i=1
where
4
y−1
−1
fδ (y) := max 3, gδ (y), gδ
+ (4cδ y) δ
+ 1.
4cδ
Proof. (i) Let n < κ. Then Corollary 2.2 yields
!
2+δ
n
X
γ
−1
2−δ/2
P
+n
P
Xi < 0 ≤ gδ
δ/2
n
i=1
n
X
!
Xi > 0 .
i=1
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
Hendrik Kläver and Norbert
Schmitz
2+δ
For the function h : [1, ∞) → [1, ∞) defined by h(y) := γ yδ/2−1 + y 2−δ/2 we
obtain
δ
δ
δ
δ
00
2+δ
−2−δ/2
h (y) =
1+
(γ
− 1)y
+ 2−
1−
y −δ/2 > 0,
2
2
2
2
i.e., h is convex. Hence the maximum of
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h : {1, . . . , bκc} → [1, ∞)
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is attained either for n = 1 or for n = bκc. Since gδ is strictly increasing this
yields
2+δ
2+δ
γ
−1
γ
−1
2−δ/2
2+δ
2+δ 4/δ−1
gδ
+n
≤ max gδ (γ ), gδ
+ (4cδ γ )
nδ/2
4cδ γ 2+δ
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Page 13 of 25
J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
for all n < κ.
(ii) Let n ≥ κ, i.e. nδ/2 ≥ 4cδ γ 2+δ . Then the (special case of the) theorem of
Berry-Esseen yields
!
n
X
γ 2+δ
1
1 Xi < 0 − ≤ cδ δ/2 ≤
P
2
n
4
i=1
and, therefore,
P
n
X
i=1
!,
Xi < 0
P
n
X
!
Xi > 0
i=1
Combining (i) and (ii) we obtain the assertion.
≤
3/4
= 3.
1/4
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
Hendrik Kläver and Norbert
Schmitz
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
3.
Some Futher Inequalities
The next inequality represents a quantification of Lemma 7 by Landers and
Rogge [8].
Lemma 3.1. Under Assumption (M)
P
min
p≤n≤q
n
X
!
Xi ≤ r
i=1
−P
max
n
X
p≤n≤q
≤ fδ (γ 2+δ ) P
!
Xi ≤ r
i=1
p
X
Xi ≤ r,
i=1
q
X
Xi ≥ r
i=1
p
+P
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
!
X
i=1
Xi ≥ r,
q
X
!!
Xi ≤ r
i=1
for all p, q ∈N s.t. p < q and for all r ∈ R.
Proof. Using the same notation (A, α, β, Ak ) as Landers and Rogge [8, p. 280],
we have to prove that
P (A) ≤ fδ (γ 2+δ )(α + β).
P
(i) First we show that P (A ∩ { pi=1 Xi ≤ r}) ≤ fδ (γ 2+δ ) · α; for this it is
sufficient to prove that
( p
) ( q
)!
X
X
P A∩
Xi ≤ r ∩
Xi ≤ r
≤ (fδ (γ 2+δ ) − 1) · α.
i=1
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
P
But due to the independence of the Ak and qi=k+1 Xi
( p
) ( q
)!
X
X
P A∩
Xi ≤ r ∩
Xi ≤ r
i=1
q−1
≤
X
i=1
q
P (Ak )P
k=p+1
!
X
Xi ≤ 0
i=k+1
q−1
≤ (fδ (γ 2+δ ) − 1)
X
P (Ak )P
k=p+1
q
X
!
Xi ≥ 0
i=k+1
according to Corollary 2.3
Hendrik Kläver and Norbert
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≤ (fδ (γ 2+δ ) − 1) · α.
(ii) Similarly, it follows that
( k
)!
X
P A∩
Xi > r
≤ fδ (γ 2+δ ) · β.
i=1
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(i) and (ii) yield the assertion.
A thorough examination of the proof of Lemma 8 of Landers and Rogge [8]
allows a generalization and quantification of their result:
Lemma 3.2. Under Assumption (M),
P
Pn+k
n
(i) P
X
≤
t,
X
≥
t
≤
i
i
i=1
i=1
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γ 2+δ
2cδ
nδ/2
+
√1
2π
q
k
.
n
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
(ii) P
P
n
i=1
Xi ≥ t,
Pn+k
i=1
Xi ≤ t ≤
2cδ γ 2+δ
nδ/2
+
√1
2π
q
k
.
n
Proof. (ii) follows from (i) by replacing Xi by −Xi . Analogously to the proof
of [8], Lemma 8, we obtain
!
n+k
n
X
X
P
Xi ≤ t,
Xi ≥ t
i=1
i=1
!
Z k
1 X
t
t
xi Q(dx1 , . . . , dxk )
−Φ √ − √
Φ √
n i=1
n
n
r Z k
k 1 X 1
xi Q(dx1 , . . . , dxk )
+√
√
2π n k i=1 r
1
k
+√
,
2π n
2cδ γ 2+δ
≤
+
nδ/2
since
≤
2cδ γ 2+δ
nδ/2
≤
2cδ γ 2+δ
nδ/2
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v
2
u k
k
1 X u 1 X
t E √
Xi ≤ E √
Xi = 1.
k
k
i=1
i=1
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4.
A Berry-Esseen Theorem for Random Sums
As an important application of these inequalities we state our main result:
Theorem 4.1 (Berry-Esseen type result for random sums). Let {Xn , n ≥
1} be i.i.d. random variables with EXn = 0, Var Xn = 1 and γ 2+δ :=
E|Xn |2+δ < ∞ for some δ ∈ (0, 1]; let {Nn , n ∈ N} be integer-valued random
variables and {ζn , n ∈ N} real numbers with limn→∞ ζn = 0 such that there
exist d, τ > 0 with
p
Nn
− 1 > ζn ≤ d ζn .
P nτ
1
Then
!
Nn
1 X
(i) sup P √
Xi ≤ t − Φ(t)
nτ
t∈R An Inequality for the
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Random Summation
Hendrik Kläver and Norbert
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i=1
1
1
+p
δ/2
bnτ c
2πebnτ c
s
p
2 + 1/τ
1
2+δ
+ 2fδ (γ )
max
, ζn + 2d ζn .
π
n
≤ cδ γ 2+δ (1 + 22+δ/2 fδ (γ 2+δ ))
(ii) sup P
t∈R !
Nn
1 X
√
Xi ≤ t − Φ(t)
Nn i=1
≤ cδ γ 2+δ (1 + 22+δ/2 fδ (γ 2+δ ))
1
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1
+p
δ/2
bnτ c
2πebnτ c
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bxc := max{n ∈ N : n ≤ x}.
J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
s
+ 2fδ (γ
for all n ∈ N s.t.
Since
nτ
2
nτ
2
2+δ
)
2 + 1/τ
max
π
1
, ζn
n
+ (3d + 1)
p
ζn
− nτ ζn ≥ 1.
−nτ ζn −→ ∞ there exists an n0 s.t.
n→∞
nτ
2
−nτ ζn ≥ 1 for all n ≥ n0 .
Proof. (i) Let n ∈ N fulfill nτ
− nτ ζn ≥ 1 and define as Landers and Rogge [8,
2
p. 271]
√
bn (t) := t nτ and In := {k ∈ N : bnτ − nτ ζn c ≤ k ≤ bnτ + nτ ζn c}.
An Inequality for the
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Random Summation
Hendrik Kläver and Norbert
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Due to the assumption on Nn we have
p
P (Nn ∈
/ In ) ≤ d ζn .
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For
(
An (t) :=
max
k∈In
k
X
)
Xi ≤ bn (t) ,
i=1
(
Bn (t) :=
min
k∈In
k
X
)
Xi ≤ bn (t)
i=1
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follows (see Landers and Rogge [8, p. 272]) for each t ∈ R
Close
P (An (t) ∩ {Nn ∈ In }) ≤ P
Nn
X
i=1
!
Xi ≤ bn (t), Nn ∈ In
≤ P (Bn (t))
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
and

bnτ c
P (An (t)) ≤ P 
X

Xi ≤ bn (t) ≤ P (Bn (t)).
i=1
Using the Berry-Esseen theorem and the result (3.3) of Petrov [10, p. 114], we
obtain


bnτ c
X


sup P
Xi ≤ bn (t) − Φ(t)
t∈R i=1


r
r
bnτ c
X
nτ 
nτ 1
Xi ≤ t
−Φ t
≤ sup P  p
bnτ c
bnτ c t∈R bnτ c i=1
r
nτ
+ sup Φ t
− Φ(t)
bnτ c
t∈R
2+δ
cδ γ
1
≤
+p
.
δ/2
bnτ c
2πebnτ c
For p(n) := bnτ − nτ ζn c, q(n) := bnτ + nτ ζn c we obtain from Lemma 3.1
 
≤ fδ (γ
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P (Bn (t)) − P (An (t))
2+δ
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
) · P 
p(n)
X
i=1
Xi ≤ bn (t) ≤
q(n)
X
i=1

Xi 
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006


p(n)
q(n)
X
X
+P 
Xi ≥ bn (t) ≥
Xi  .
i=1
i=1
According to Lemma 3.2 it follows that


q(n)
p(n)
X
X
sup P 
Xi ≤ bn (t) ≤
Xi 
t∈R
i=1
i=1
2+δ
s
1
2cδ γ
q(n) − p(n)
+√
δ/2
(p(n))
p(n)
2π
s
1
21+δ/2 cδ γ 2+δ
2 + 1/τ
≤
+
max
, ζn ,
(nτ )δ/2
π
n
≤
since p(n) ≥ nτ − nτ ζn − 1 ≥ nτ /2 and, therefore,
s
s
r
q(n) − p(n)
2nτ ζn + 1
2
≤
= 4ζn +
;
p(n)
nτ /2
nτ
analogously


s
q(n)
p(n)
1+δ/2
2+δ
X
X
2
c
γ
2
+
1/τ
1
δ
sup P 
Xi ≥ bn (t) ≥
Xi  ≤
+
max
, ζn .
δ/2
(nτ
)
π
n
t∈T
i=1
i=1
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
Altogether we obtain
!
Nn
X
sup P
Xi ≤ bn (t) − Φ(t)
t∈R
i=1
!
Nn
X
≤ sup P
Xi ≤ bn (t), Nn ∈ In − Φ(t) + P (Nn ∈
/ In )
t∈R
i=1
p
1
cδ γ 2+δ
p
+
+
2d
ζn
bnτ cδ/2
2πebnτ c
s
!
1+δ
2+δ
2
c
γ
1
2
+
1/τ
δ
≤ 2fδ γ 2+δ
+
max
, ζn
(nτ )δ/2
π
n
p
1
cδ γ 2+δ
p
+
+
+
2d
ζn
bnτ cδ/2
2πebnτ c
1
1
≤ cδ γ 2+δ (1 + 22+δ/2 fδ (γ 2+δ ))
+p
δ/2
bnτ c
2πebnτ c
s
p
2 + 1/τ
1
2+δ
+ 2fδ (γ )
max
, ζn + 2d ζn .
π
n
≤ P (Bn (t)) − P (An (t)) +
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
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(ii) Applying Lemma 1 of Michel and Pfanzagl [9] for
Close
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Nn
1 X
f=√
Xi ,
nτ i=1
r
g=
Nn
nτ
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J. Ineq. Pure and Appl. Math. 7(1) Art. 2, 2006
q
√
and using the fact that Nnτn − 1 > ζn implies Nnτn − 1 > ζn , hence
P
r
!
N
p
p
Nn
n
− 1 > ζn ≤ P − 1 > ζn ≤ d ζn ,
nτ
nτ
we obtain from part (i)
sup P
t∈T 1
√
Nn
Nn
X
i=1
!
Xi ≤ t
− Φ(t)
1
1
p
+
bnτ cδ/2
2πebnτ c
s
p
1
2 + 1/τ
2+δ
+ 2fδ (γ )
max
, ζn + (3d + 1) ζn .
π
n
≤ cδ γ 2+δ (1 + 22+δ/2 fδ (γ 2+δ ))
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
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References
[1] F.J. ANSCOMBE, Large sample theory of sequential estimation, Proc.
Cambridge Philos. Soc., 48 (1952), 600–607.
[2] P. VAN BEEK, An application of Fourier methods to the problem of sharpening the Berry-Esseen inequality, Z. Wahrscheinlichkeitstheorie verw.
Gebiete, 23 (1972), 187–196.
[3] J.R. BLUM, D.L. HANSON AND J.I. ROSENBLATT, On the central limit
theorem for the sum of a random number of independent random variables,. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 1 (1963), 389–393.
An Inequality for the
Asymmetry of Distributions and
a Berry-Esseen Theorem for
Random Summation
[4] H. CALLAERT AND P. JANSSEN, A note on the convergence rate of
random sums, Rev. Roum. Math. Pures et Appl., 28 (1983), 147–151.
Hendrik Kläver and Norbert
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[5] Y.S. CHOW AND H. TEICHER, Probability Theory: Independence, Interchangeability, Martingales. Springer, New York, 1978.
Title Page
[6] H. KLÄVER, Ein Berry-Esseen-Satz für Zufallssummen. Diploma Thesis,
Univ. Münster, 2002
[7] A. KRAJKA AND Z. RYCHLIK, The order of approximation in the central limit theorem for random summation, Acta Math. Hungar., 51 (1988),
109–115.
[8] D. LANDERS AND L. ROGGE, The exact approximation order in the central limit theorem for random summation, Z. Wahrscheinlichkeitstheorie
verw. Gebiete, 36 (1976), 269–283.
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[9] R. MICHEL AND J. PFANZAGL, The accuracy of the normal approximations for minimum contrast estimates, Z. Wahrscheinlichkeitstheorie verw.
Gebiete, 18 (1971), 73–84.
[10] V.V. PETROV, Sums of Independent Random Variables. Springer, Heidelberg, 1975.
[11] A. RENYI, On the central limit theorem for the sum of a random number
of independent random variables. Acta Math. Acad. Sci. Hung., 11 (1960),
97–102.
[12] W. TYSIAK, Gleichmäßige Berry-Esseen-Abschätzungen. Ph. Thesis,
Univ. Wuppertal, 1983.
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Asymmetry of Distributions and
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Random Summation
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