Journal of Inequalities and Applications Volume 2009, Article ID 957056,10pages 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,[email protected]
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 andXis in the domain of attraction of the normal law andEX0. For 1≤p <2, b >−1, we prove the precise asymptotics in Davis law of large numbers for∞n1lognb/nE{|Sn|/Vn−
ε2 logn2−p/2p}asε0.
Copyrightq2009 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 ofi.i.d.random variables andXis
in the domain of attraction of the normal law andEX0. Put
Sn n
k1
Xk, Vn2 n
i1
X2i. 1.1
Also let lognlnn∨e.Then by the well-known Davis laws of large numbers1,
∞
n1 logn
n P
|Sn| ≥ε
nlogn
<∞, ε >0, 1.2
Gut and Sp˘ataru2proved its precise asymptotics as follows.
Theorem A. Suppose thatEX10andEX21 σ2<∞.Then for0≤δ≤1,
lim
ε0ε
2δ1∞
n1
lognδ
n P
|Sn| ≥ε
nlogn
μ2δ2
δ1 σ
2δ2, 1.3
whereμ2δ2stands for the2δ2thabsolute moment of the standard normal distribution.
It is well known that, fori.i.d. random variables, Chow3discussed the complete moment convergence, and got the following result.
Theorem B. Let{X, Xn;n ≥ 1}be a sequence ofi.i.d.random variables with EX1 0 . Assume
p≥1,α >1/2, pα >1, andE|X|p|X|log1|X|<∞.Then for anyε >0,
∞
n1
npα−2−αE max
j≤n |Sj| −εn α
<∞. 1.4
On the other hand, the past decade has witnessed a significant development on the limit theorems for the so-called self-normalized sumSn/Vn, Vn
n
i1Xi2. Bentkus and
G ¨otze 4obtained Berry-Esseen inequalities for self-normalized sums. Wang and Jing 5
derived exponential nonuniform Berry-Esseen bound. Gin´e et al.6, established asymptotic normality of self-normalized sums.
Theorem C. Let{X, Xn;n≥1}be a sequence ofi.i.d.random variables withEX1 0. Then for any
x∈R,
lim
n→ ∞P
Sn
Vn ≤x
Φ x 1.5
holds, if and only ifXis in the domain of attraction of the normal law, whereΦxis the distribution function of the standard normal random variable.
Shao 7 showed a self-normalization large deviation result for PSn/Vn ≥ x√n
without any moment conditions.
Theorem D. Let{xn;n≥1}be a sequence of positive numbers withxn → ∞andxn o√nas
n → ∞. IfEX0andEX2I|X| ≤xis slowly varying asx → ∞,then
lim
n→ ∞x −2
n lnP
Sn
Vn ≥xn
−1
2. 1.6
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. SupposeX is in the domain of attraction of the normal law and EX 0. Then, for b >−1and1≤p <2, one has
lim
ε0ε
2pb1/2−p∞
n1
lognb
n E
|Sn|
Vn −ε
2 logn2−p/2p
2−b−1
2−p
b12pbp2E|N|
2pbp2/2−p,
1.7
here and in the sequel,Nis the standard normal random variable.
Remark 1.2. Ifp 1 and 0 < σ2 EX2 < ∞, by the strong law of large numbers, we have V2
n/n → σ2, a.s.Then, we can easily obtain the following result:
lim
ε0ε
2b1∞
n1
lognb
n3/2 E |Sn| −εσ
2nlogn
σ2−b−1
b1 2b3E|N|
2b3. 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 setAε expM/ε2p/2−p,forM >1 andε >0. Here and in the sequel,
Cwill denote positive constants, possibly varying from place to place, andx means the largest integer≤x. The proof ofTheorem 1.1is based on the following propositions.
Proposition 2.1. Forb >−1, one has
lim
ε0ε
2pb1/2−p∞
n1
lognb
n E
|N| −ε2 logn2−p/2p
2−b−1
2−p
b12pbp2E|N|
2pbp2/2−p.
Proof. Via the change of variableyε2 logt2−p/2p, we have
lim
ε0ε
2pb1/2−p∞
n1
lognb
n E
|N| −ε2 logn2−p/2p
lim
ε0ε
2pb1/2−p∞
n1
lognb n
∞
ε2 logn2−p/2p
P|N| ≥xdx
lim
ε0ε
2pb1/2−p∞
e
logtb
t
∞
ε2 logt2−p/2p
P|N| ≥xdxdt
lim
ε0
p2−b
2−p
∞
ε22−p/2py
2p/2−pb1−1 ∞
y
P|N| ≥xdxdy
lim
ε0
p2−b
2−p
∞
ε22−p/2pP|N| ≥x
x
ε22−p/2py
2p/2−pb1−1dydx
lim
ε0 2−b−1
b1 ∞
ε22−p/2pP|N| ≥x
x2p/2−pb1−ε2p/2−pb1·2b1dx
lim
ε0 2−b−1
b1 ∞
ε22−p/2px
2p/2−pb1P|N| ≥xdx
2−b−1
2−p
b12pbp2E|N|
2pbp2/2−p.
2.2
Proposition 2.2. Forb >−1, one has
lim
ε0ε
2pb1/2−p n≤Aε
lognb
n
E |Sn|
Vn −ε
2 logn2−p/2p
−E
|N|−ε2 logn2−p/2p
0.
2.3
Proof. SetΔnsupx∈R|P|Sn|/Vn≥x−P|N| ≥x|.Then, by1.5, it is easy to seeΔn → 0
asn → ∞.Observe that
lim
ε0ε
2pb1/2−p
n≤Aε
lognb
n
E |Sn|
Vn −ε
2 logn2−p/2p
−E
|N| −ε2 logn2−p/2p
lim
ε0ε
2pb1/2−p n≤Aε
lognb
n × ∞ 0 P |
Sn|
Vn ≥xε
2 logn2−p/2p
dx−
∞
0
≤lim
ε0ε
2pb1/2−p
n≤Aε
lognb
n
∞
0 P
|
Sn|
Vn ≥xε
2 logn2−p/2p
− ∞
0
P|N| ≥xε2 logn2−p/2pdx
≤lim
ε0ε
2pb1/2−p
n≤Aε
lognb
n Δn1 Δn2 Δn3 Δn4, 2.4
where
Δn1
minlogn,1/√Δn
0
P
|
Sn|
Vn ≥xε
2 logn2−p/2p
−P|N| ≥xε2 logn2−p/2pdx,
Δn2 n1/4
minlogn,1/√Δn
P
|
Sn|
Vn ≥xε
2 logn2−p/2p
−P|N| ≥xε2 logn2−p/2pdx,
Δn3 n1/2
n1/4 P
|
Sn|
Vn ≥xε
2 logn2−p/2p
−P|N| ≥xε2 logn2−p/2pdx,
Δn4 ∞
n1/2 P|Sn|
Vn ≥xε
2 logn2−p/2p−P|N| ≥xε2 logn2−p/2pdx.
2.5
Thus forΔn1, it is easy to see
Δn1≤
Δn−→0, asn−→ ∞. 2.6
Now we are in a position to estimateΔn2. From 1.6, and by applying−Xisto it, we can obtain that for large enoughnand any 0< a≤1/4, there exist C and b such thatP|Sn|/Vn>
x≤Ce−1/2−ax2
forb < x < n1/2/b. In particular, forb < x < n1/2/b, there existsC >0 such that
P
|
Sn|
Vn > x
Hence, by Markov’s inequality and2.7, we have
Δn2≤ n1/4
minlogn,1/√Δne
−xε2 logn2−p/2p2/4
dx
n1/4
minlogn,1/√Δn
C
xε2 logn2−p/2p2 dx
≤ n1/4
minlogn,1/√Δn
e−x2/4dx
n1/4
minlogn,1/√Δn
C
x2dx−→0, asn −→ ∞.
2.8
ForΔn3, by Markov’s inequality and2.7, we have
Δn3≤ n1/2
n1/4P |
Sn|
Vn ≥n
1/4
dx
n1/2
n1/4
C
xε2 logn2−p/2p2 dx
≤e−√n/4n1/2−n1/4 n1/2
n1/4
C
x2dx−→0, asn−→ ∞.
2.9
From Cauchy inequality, it follows that
|Sn|
Vn ≤
√
n. 2.10
Therefore
Δn4 ∞
n1/2
P|N| ≥xε2 logn2−p/2pdx
≤ ∞
n1/2
C
xε2 logn2−p/2p2 dx
≤ ∞
n1/2
C
x2dx−→0, asn−→ ∞.
2.11
DenoteΔn Δn1 Δn2 Δn3 Δn4, then, since the weighted average of a sequence that converges to 0 also converges to 0, it follows that, for anyM >1,
lim
ε0ε
2pb1/2−p
n≤Aε
lognb
n
E |Sn|
Vn −ε
2 logn2−p/2p
−E
|N| −ε2 logn2−p/2p
≤lim
ε0ε
2pb1/2−p
n≤Aε
lognb
n Δ
n−→0, asε0.
2.12
Proposition 2.3. Forb >−1, one has
lim
M→ ∞limε0ε
2pb1/2−p
n>Aε
lognb
n E
|N| −ε2 logn2−p/2p
0. 2.13
Proof. Note that
ε2pb1/2−p
n>Aε
lognb
n E
|N| −ε2 logn2−p/2p
≤ε2pb1/2−p
∞
Aε
lognb
t
∞
ε2 logt2−p/2pP|N| ≥xdx dt
≤ ∞
√
2M
y2p/2−pb1−1
∞
y
P|N| ≥xdx dy
∞
√
2M
P|N| ≥x
x
√
2M
y2p/2−pb1−1dy dx
≤C
∞
√
2M
x2p/2−pb1P|N| ≥xdx−→0, asM−→ ∞.
2.14
So this proposition is proved now.
Proposition 2.4. Forb >−1, one has
lim
M→ ∞ limε0ε
2pb1/2−p
n>Aε
lognb
n E
|Sn|
Vn −ε
2 logn2−p/2p
0. 2.15
Proof. Note that
ε2pb1/2−p n>Aε
lognb
n E
|Sn|
Vn −ε
2 logn2−p/2p
ε2pb1/2−p
n>Aε
lognb
n
∞
0
P
|
Sn|
Vn ≥xε
2 logn2−p/2p
dx
B1B2B3,
where
B1ε2pb1/2−p
n>Aε
lognb
n
n1/4
0
P
|
Sn|
Vn ≥xε
2 logn2−p/2p
dx,
B2ε2pb1/2−p
n>Aε
lognb
n
n1/2
n1/4P |
Sn|
Vn ≥xε
2 logn2−p/2p
dx,
B3ε2pb1/2−p
n>Aε
lognb
n
∞
n1/2P |
Sn|
Vn ≥xε
2 logn2−p/2p
dx.
2.17
ForB1, by2.7, we have
B1≤Cε2pb1/2−p
n>Aε
lognb
n
n1/4
0
e−xε2 logn2−p/2p2/4dx
≤Cε2pb1/2−p
n>Aε
lognb
n
∞
0
e−xε2 logn2−p/2p2/4dx
Cε2pb1/2−p
n>Aε
lognb
n
∞
ε2 logn2−p/2pe −x2/4
dx
≤Cε2pb1/2−p
∞
Aε
lognb
t
∞
ε2 logt2−p/2pe −x2/4
dxdt
≤C
∞
√
2M
y2p/2−pb1−1
∞
y
e−x2/4dxdy
C
∞
√
2M
e−x2/4
x
√
2M
y2p/2−pb1−1dydx
≤C
∞
√
2M
x2p/2−pb1e−x2/4dx−→0, asM−→ ∞.
2.18
ForB2, using2.7again, we have
B2≤ε2pb1/2−p
n>Aε
lognb
n
n1/2−n1/4P |
Sn|
Vn ≥n
1/4ε2 logn2−p/2p
≤Cε2pb1/2−p
n>Aε
lognb
n
≤Cε2pb1/2−p
n>Aε
lognb
n
n1/2−n1/4e−√n/4e−ε22 logn2−p/p/4
≤Cε2pb1/2−p
n>Aε
lognb
n e
−ε22 logn2−p/p/4
≤Cε2pb1/2−p
∞
Aε
lognb
t e
−ε22 logt2−p/p/4 dt
by lettingz ε
22 logt2−p/p 4
≤C
∞
2M2−p/p/4z
pb1/2−p−1e−zdz−→0, asM−→ ∞.
2.19
By noting that2.10, it is easily seen that
B3 0. 2.20
Combining2.18,2.19, and2.20, 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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