Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 957056, 10 pages
doi:10.1155/2009/957056
Research Article
A Limit Theorem for the Moment of
Self-Normalized Sums
Qing-pei Zang
Department of Mathematics, Huaiyin Teachers College, Huaian 223300, China
Correspondence should be addressed to Qing-pei Zang, zqphunhu@yahoo.com.cn
Received 25 December 2008; Revised 30 March 2009; Accepted 18 June 2009
Recommended by Jewgeni Dshalalow
Let {X, Xn ; n ≥ 1} be a sequence of independent and identically distributed i.i.d. random
variables and X is in the domain of attraction of the normal law and EX 0. For 1 ≤ p < 2, b > −1,
b
we prove the precise asymptotics in Davis law of large numbers for ∞
n1 log n /nE{|Sn |/Vn −
2−p/2p
} as ε 0.
ε2 log n
Copyright q 2009 Qing-pei Zang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction and Main Result
Throughout this paper, we let {X, Xn ; n ≥ 1} be a sequence of i.i.d. random variables and X is
in the domain of attraction of the normal law and EX 0. Put
Sn n
Xk ,
k1
Vn2 n
1.1
Xi2 .
i1
Also let log n lnn ∨ e. Then by the well-known Davis laws of large numbers 1,
∞
log n
P |Sn | ≥ ε n log n < ∞,
n
n1
if and only if EX 0 and EX 2 < ∞.
ε > 0,
1.2
2
Journal of Inequalities and Applications
Gut and Spătaru 2 proved its precise asymptotics as follows.
Theorem A. Suppose that EX 1 0 and EX12 σ 2 < ∞. Then for 0 ≤ δ ≤ 1,
lim ε
ε0
2δ1
∞ log n δ
μ2δ2 2δ2
P |Sn | ≥ ε n log n σ
,
n
δ1
n1
1.3
where μ2δ2 stands for the 2δ 2th absolute moment of the standard normal distribution.
It is well known that, for i.i.d. random variables, Chow 3 discussed the complete
moment convergence, and got the following result.
Theorem B. Let {X, Xn ; n ≥ 1} be a sequence of i.i.d. random variables with EX1 0 . Assume
p ≥ 1, α > 1/2, pα > 1, and E|X|p |X| log1 |X| < ∞. Then for any ε > 0,
∞
npα−2−α E max|Sj | − εnα
j≤n
n1
< ∞.
1.4
On the other hand, the past decade has witnessed a significant
development on the
n
2
limit theorems for the so-called self-normalized sum Sn /Vn , Vn i1 Xi . Bentkus and
Götze 4 obtained Berry-Esseen inequalities for self-normalized sums. Wang and Jing 5
derived exponential nonuniform Berry-Esseen bound. Giné et al. 6, established asymptotic
normality of self-normalized sums.
Theorem C. Let {X, Xn ; n ≥ 1} be a sequence of i.i.d. random variables with EX1 0 . Then for any
x ∈ R,
lim P
n→∞
Sn
≤x
Vn
Φ x
1.5
holds, if and only if X is in the domain of attraction of the normal law, where Φx is the distribution
function of the standard normal random variable.
√
Shao 7 showed a self-normalization large deviation result for P Sn /Vn ≥ x n
without any moment conditions.
√
Theorem D. Let {xn ; n ≥ 1} be a sequence of positive numbers with xn → ∞ and xn o n as
n → ∞. If EX 0 and EX 2 I|X| ≤ x is slowly varying as x → ∞, then
lim xn−2
n→∞
ln P
Sn
≥ xn
Vn
1
− .
2
1.6
Since then, many subsequent developments of self-normalized sums have been
obtained. For example, Csörgő et al. 8 have established Darling-Erdös theorem for selfnormalized sums, and they 9 have also obtained Donsker’s theorem for self-normalized
partial sums processes.
Journal of Inequalities and Applications
3
Inspired by the above results, in this note we study the precise asymptotics in Davis
law of large numbers for the moment of self-normalized sums. Our main result is as follows.
Theorem 1.1. Suppose X is in the domain of attraction of the normal law and EX 0. Then, for
b > −1 and 1 ≤ p < 2, one has
lim ε
2pb1/2−p
ε0
∞ log n b
2−p/2p
|Sn |
E
− ε 2 log n
n
V
n
n1
2−b−1 2 − p
E|N|2pbp2/2−p ,
b 1 2pb p 2
1.7
here and in the sequel, N is the standard normal random variable.
Remark 1.2. If p 1 and 0 < σ 2 EX 2 < ∞, by the strong law of large numbers, we have
Vn2 /n → σ 2 , a.s. Then, we can easily obtain the following result:
lim ε2b1
ε0
∞ log n b
n1
n3/2
E |Sn | − εσ
2n log n
σ2−b−1
E|N|2b3 .
b 1 2b 3
1.8
Remark 1.3. As is well known, the strong approximation method is taken in order to obtain
such an analogous result, however, this method is not applicable here.
2. Proof of Theorem 1.1
In this section, we set Aε expM/ε2p/2−p , for M > 1 and ε > 0. Here and in the sequel,
C will denote positive constants, possibly varying from place to place, and x means the
largest integer ≤ x. The proof of Theorem 1.1 is based on the following propositions.
Proposition 2.1. For b > −1, one has
lim ε
ε0
2pb1/2−p
∞ log n b
2−p/2p
E |N| − ε 2 log n
n
n1
2−b−1 2 − p
E|N|2pbp2/2−p .
b 1 2pb p 2
2.1
4
Journal of Inequalities and Applications
Proof. Via the change of variable y ε2 log t2−p/2p , we have
lim ε
2pb1/2−p
ε0
∞ log n b
2−p/2p
E |N| − ε 2 log n
n
n1
lim ε
2pb1/2−p
lim ε
2pb1/2−p
ε0
∞ log n b
n
n1
∞
ε0
e
p2−b
ε0 2 − p
∞
−b
∞
lim
ε22−p/2p
p2
ε0 2 − p
lim
ε22−p/2p
2−b−1
ε0 b 1
lim
b
log t
t
∞
2−p/2p
ε2 log n
∞
ε2 log t
y2p/2−pb1−1
2−p/2p
∞
P |N| ≥ x dx
P |N| ≥ x dxdt
P |N| ≥ x dxdy
y
P |N| ≥ x
∞
ε22−p/2p
x
ε22−p/2p
2.2
y2p/2−pb1−1 dydx
P |N| ≥ x x2p/2−pb1 − ε2p/2−pb1 · 2b1 dx
2−b−1 ∞
x2p/2−pb1 P |N| ≥ x dx
ε0 b 1 ε22−p/2p
2−b−1 2 − p
E|N|2pbp2/2−p .
b 1 2pb p 2
lim
Proposition 2.2. For b > −1, one has
log n b
n
n≤Aε
lim ε2pb1/2−p
ε0
E |Sn | −ε 2 log n 2−p/2p
Vn
2−p/2p
−E |N|−ε 2 log n
0.
2.3
Proof. Set Δn supx∈R |P |Sn |/Vn ≥ x − P |N| ≥ x|. Then, by 1.5, it is easy to see Δn → 0
as n → ∞. Observe that
lim ε
2pb1/2−p
ε0
n≤Aε
b
log n
n
lim ε2pb1/2−p
ε0
× n≤Aε
∞
P
0
E |Sn | − ε 2 log n 2−p/2p
Vn
2−p/2p
− E |N| − ε 2 log n
b
log n
n
2−p/2p
|Sn |
≥ x ε 2 log n
dx −
Vn
∞
0
2−p/2p dx
P |N| ≥ x ε 2 log n
Journal of Inequalities and Applications
≤ lim ε
2pb1/2−p
ε0
b
log n
n
n≤Aε
∞
0
5
P |Sn | ≥ x ε 2 log n 2−p/2p
V
n
−
∞
0
≤ lim ε2pb1/2−p
ε0
2−p/2p dx
P |N| ≥ x ε 2 log n
n≤Aε
b
log n
Δn1 Δn2 Δn3 Δn4 ,
n
2.4
where
Δn1 minlog n,1/
√
Δn n
0
Δn2 2−p/2p − P |N| ≥ x ε 2 log n
dx,
n1/4
minlog n,1/
Δn3 √
Δn P |Sn | ≥ x ε 2 log n 2−p/2p
V
n
2−p/2p − P |N| ≥ x ε 2 log n
dx,
2.5
P |Sn | ≥ x ε 2 log n 2−p/2p − P |N| ≥ x ε 2 log n 2−p/2p dx,
V
1/4
n1/2
n
n
Δn4 P |Sn | ≥ x ε 2 log n 2−p/2p
V
|Sn |
2−p/2p
2−p/2p dx.
P
≥
x
ε
2
log
n
−
P
≥
x
ε
2
log
n
|N|
V
1/2
∞
n
n
Thus for Δn1 , it is easy to see
Δn1 ≤
Δn −→ 0,
as n −→ ∞.
2.6
Now we are in a position to estimate Δn2 . From 1.6, and by applying −Xi s to it, we can
obtain that for large enough n and any 0 < a ≤ 1/4, there exist C and b such that P |Sn |/Vn >
2
x ≤ Ce−1/2−ax for b < x < n1/2 /b. In particular, for b < x < n1/2 /b, there exists C > 0 such
that
P
|Sn |
>x
Vn
≤ Ce−x
2
/4
.
2.7
6
Journal of Inequalities and Applications
Hence, by Markov’s inequality and 2.7, we have
Δn2 ≤
n1/4
minlog n,1/
≤
√
Δn e
n1/4
minlog n,1/
√
Δn e
−xε2 log n2−p/2p 2 /4
−x2 /4
dx n1/4
dx √
minlog n,1/ Δn x
n1/4
C
2
√
minlog n,1/ Δn x
dx −→ 0,
C
2−p/2p 2 dx
ε 2 log n
as n −→ ∞.
2.8
For Δn3 , by Markov’s inequality and 2.7, we have
Δn3 ≤
n1/2
P
n1/4
≤ e−
√
n/4
|Sn |
≥ n1/4 dx Vn
n1/2 − n1/4 n1/2
n1/4
1/2
n
n1/4
C
2−p/2p 2 dx
x ε 2 log n
C
dx −→ 0,
x2
2.9
as n −→ ∞.
From Cauchy inequality, it follows that
|Sn | √
≤ n.
Vn
2.10
Therefore
∞
Δn4 n1/2
2−p/2p dx
P |N| ≥ x ε 2 log n
∞
≤
n1/2
∞
C
dx −→ 0,
2
n1/2 x
≤
C
2−p/2p 2 dx
x ε 2 log n
2.11
as n −→ ∞.
Denote Δn Δn1 Δn2 Δn3 Δn4 , then, since the weighted average of a sequence that
converges to 0 also converges to 0, it follows that, for any M > 1,
lim ε
2pb1/2−p
ε0
n≤Aε
≤ lim ε
ε0
b
log n
n
2pb1/2−p
n≤Aε
E |Sn | − ε 2 log n 2−p/2p
Vn
b
log n
Δn −→ 0,
n
2−p/2p
− E |N| − ε 2 log n
as ε 0.
2.12
The proof is completed.
Journal of Inequalities and Applications
7
Proposition 2.3. For b > −1, one has
lim lim ε
2pb1/2−p
M → ∞ ε0
b
log n
2−p/2p
E |N| − ε 2 log n
n
n>Aε
0.
2.13
Proof. Note that
ε2pb1/2−p
n>Aε
b
log n
2−p/2p
E |N| − ε 2 log n
n
≤ε
2pb1/2−p
∞
Aε
≤
∞
√
2M
∞
√
2M
≤C
b
log n
t
y2p/2−pb1−1
∞
∞
ε2 log t2−p/2p
P |N| ≥ x dx dt
2.14
P |N| ≥ x dx dy
y
P |N| ≥ x
∞
x
√
2M
x
y
√
2M
2p/2−pb1
2p/2−pb1−1
dy dx
P |N| ≥ x dx −→ 0,
as M −→ ∞.
So this proposition is proved now.
Proposition 2.4. For b > −1, one has
lim lim ε
2pb1/2−p
M → ∞ ε0
n>Aε
b
log n
2−p/2p
|Sn |
E
− ε 2 log n
n
Vn
0.
2.15
Proof. Note that
ε
2pb1/2−p
n>Aε
ε
b
log n
2−p/2p
|Sn |
E
− ε 2 log n
n
Vn
2pb1/2−p
n>Aε
B1 B 2 B 3 ,
b
log n
n
∞
P
0
2−p/2p
|Sn |
≥ x ε 2 log n
dx
Vn
2.16
8
Journal of Inequalities and Applications
where
B1 ε
b
log n
n
2pb1/2−p
n>Aε
B2 ε
B3 ε
n1/2
2−p/2p
|Sn |
≥ x ε 2 log n
dx,
Vn
n1/4
b
log n
n
n>Aε
P
2pb1/2−p
2−p/2p
|Sn |
≥ x ε 2 log n
dx,
Vn
0
b
log n
n
n>Aε
P
2pb1/2−p
n1/4
∞
P
n1/2
2.17
2−p/2p
|Sn |
≥ x ε 2 log n
dx.
Vn
For B1 , by 2.7, we have
B1 ≤ Cε2pb1/2−p
b
log n
n
n>Aε
≤ Cε2pb1/2−p
b
log n
n
Cε
≤ Cε
∞
Aε
∞
≤C
√
√
√
∞
dx
2−p/2p 2
e−xε2 log n
/4
dx
∞
ε2 log n
2−p/2p
∞
ε2 log t2−p/2p
e−x
2
/4
e−x
e−x
2
2
/4
/4
dx
2.18
dxdt
dxdy
y
2M
e−x
2
/4
2M
∞
≤C
b
log n
t
y2p/2−pb1−1
∞
C
/4
0
b
log n
n
n>Aε
2pb1/2−p
∞
2−p/2p 2
e−xε2 log n
0
n>Aε
2pb1/2−p
n1/4
x
√
y2p/2−pb1−1 dydx
2M
x2p/2−pb1 e−x
2
/4
dx −→ 0,
as M −→ ∞.
2M
For B2 , using 2.7 again, we have
B2 ≤ ε2pb1/2−p
n>Aε
≤ Cε
2pb1/2−p
b
|S |
log n 1/2
2−p/2p
n
n − n1/4 P
≥ n1/4 ε 2 log n
n
Vn
n>Aε
b
log n 1/2
2−p/2p 2
1/4
/4
n − n1/4 e−n ε2 log n
n
Journal of Inequalities and Applications
≤ Cε
2pb1/2−p
b
√
log n 1/2
2−p/p
2
/4
n − n1/4 e− n/4 e−ε 2 log n
n
n>Aε
≤ Cε
2pb1/2−p
b
log n
2−p/p
2
/4
e−ε 2 log n
n
n>Aε
≤ Cε2pb1/2−p
∞
Aε
9
b
log n
2−p/p
2
/4
e−ε 2 log t
dt
t
2−p/p ε2 2 log t
by letting z 4
≤C
∞
2M
2−p/p
zpb1/2−p−1 e−z dz −→ 0,
as M −→ ∞.
/4
2.19
By noting that 2.10, it is easily seen that
B3 0.
2.20
Combining 2.18, 2.19, and 2.20, the proposition is proved.
Our main result follows from the propositions using the triangle inequality.
Acknowledgments
The author thanks the referees for pointing out some errors in a previous version, as well
as for several comments that have led to improvements in this work. Thanks are also due
to Doctor Ke-ang Fu of Zhejiang University in china for his valuable suggestion in the
preparation of this paper.
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Journal of Inequalities and Applications
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