Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 329391, 13 pages
doi:10.1155/2012/329391
Research Article
Almost Sure Central Limit Theory for
Self-Normalized Products of Sums of Partial Sums
Qunying Wu1, 2
1
2
College of Science, Guilin University of Technology, Guilin 541004, China
Guangxi Key Laboratory of Spatial Information and Geomatics, Guilin University of Technology,
Guilin 541004, China
Correspondence should be addressed to Qunying Wu, wqy666@glite.edu.cn
Received 18 January 2012; Accepted 7 February 2012
Academic Editor: Naseer Shahzad
Copyright q 2012 Qunying Wu. 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.
Let X, X1 , X2 , . . . be a sequence of independent and identically distributed random variables in
the domain of attraction of a normal law. An almost sure limit theorem for the self-normalized
products of sums of partial sums is established.
1. Introduction
Let {X, Xn }n∈N be a sequence of independent and identically distributed i.i.d. positive
random variables with a nondegenerate distribution function and EX μ > 0. For each
n
n ≥ 1, the symbol Sn /Vn denotes self-normalized partial sums, where Sn i1 Xi and
Vn2 ni1 Xi − μ2 . We say that the random variable X belongs to the domain of attraction of
the normal law if there exist constants an > 0, bn ∈ R such that
Sn − bn d
−→ N,
an
as n −→ ∞,
1.1
here and in the sequel N is a standard normal random variable. We say that {X, Xn }n∈N satisfies the central limit theorem CLT.
It is known that 1.1 holds if and only if
x2 P|X| > x
0.
x → ∞ EX 2 I|X| ≤ x
lim
1.2
2
Journal of Applied Mathematics
In contrast to the well-known classical central limit theorem, Gine et al. 1 obtained the
d
following self-normalized version of the central limit theorem: Sn −ESn /Vn → N as n → ∞
if and only if 1.2 holds.
Brosamler 2 and Schatte 3 obtained the following almost sure central limit theorem
ASCLT: let {Xn }n∈N be i.i.d. random variables with mean 0, variance σ 2 > 0, and partial
sums Sn . Then
n
1 Sk
dk I
√ < x Φx
n → ∞ Dn
σ k
k1
lim
a.s. ∀x ∈ R,
1.3
with dk 1/k and Dn nk1 dk ; here and in the sequel I denotes an indicator function, and
Φx is the standard normal distribution function. Some ASCLT results for partial sums were
obtained by Lacey and Philipp 4, Ibragimov and Lifshits 5, Miao 6, Berkes and Csáki
7, Hörmann 8, Wu 9, 10, and Ye and Wu 11. Huang and Pang 12 and Zhang and
Yang 13 obtained ASCLT results for self-normalized version. However, ASCLT results for
self-normalized products of sums of partial sums have not been reported yet.
Under mild moment conditions, ASCLT follows from the ordinary CLT, but in general
the validity of ASCLT is a delicate question of a totally different character as CLT. The
difference between CLT and ASCLT lies in the weight in ASCLT. The terminology of
summation procedures see e.g., Chandrasekharan and Minakshisundaram 14, page 35
shows that the larger the weight sequence {dk ; k ≥ 1} in 1.3 is, the stronger the relation
becomes. By this argument, one should also expect to get stronger results if we use larger
weights. And it would be of considerable interest to determine the optimal weights.
On the other hand, by the Theorem 1 of Schatte 3, 1.3 fails for weight dk 1. The
optimal weight sequence remains unknown.
The purpose of this paper is to study and establish the ASCLT for self-normalized
products of sums of partial sums of random variables in the domain of attraction of the
normal law, we will show that the ASCLT holds under a fairly general growth condition
on dk k−1 expln kα , 0 ≤ α < 1/2.
In the following, we assume that {X, Xn }n∈N is a sequence of i.i.d. positive random
variables in the domain of attraction of the normal law with EX μ > 0 and define Sk k
k
n
n
k
2
i1 Xi , Vk i1 Xi − μ , Tk i1 Si . Let bn,k jk 1/j, cn,k 2
jk j 1 − k/jj 1,
and dn,k n 1 − k/n 1 for 1 ≤ k ≤ n. IA denotes the indicator function of set A,
and an ∼ bn denotes an /bn → 1, n → ∞. The symbol c stands for a generic positive constant,
which may differ from one place to another. Let
2 b inf{x ≥ 1; lx > 0},
lx E X − μ I X − μ ≤ x ,
ls 1
ηj inf s; s ≥ b 1, 2 ≤
for j ≥ 1.
j
s
1.4
By the definition of ηj , we have jlηj ≤ ηj2 and jlηj − ε > ηj − ε2 for any ε > 0. It implies that
nl ηn ∼ ηn2 ,
as n → ∞, ηn < n 1.
Our theorem is formulated in a more general setting.
1.5
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3
Theorem 1.1. Let {X, Xn }n∈N be a sequence of i.i.d. positive random variables in the domain of
attraction of the normal law with EX μ > 0. Suppose
l ηn ∼ l
ηn
.
ln n
1.6
For 0 ≤ α < 1/2, set
exp lnα k
,
dk k
Dn n
1.7
dk .
k1
Then
⎛⎛
⎞
⎞μ/Vk
k
n
T
1
j
⎜
⎟
⎠
dk I ⎝ ⎝
≤ x⎠ Fx
lim
n → ∞ Dn
j
j
1
μ/2
j1
k1
a.s.,
1.8
for any x ∈ R, where F is the distribution function of the random variable exp 10/3N.
By the terminology of summation procedures, we have the following corollary.
Corollary 1.2. Theorem 1.1 remains valid if we replace the weight sequence {dk }k∈N by {dk∗ }k∈N such
∗
that 0 ≤ dk∗ ≤ dk , ∞
k1 dk ∞.
Remark 1.3. If EX 2 σ 2 < ∞, then X is in the domain of attraction of the normal law and lx →
σ 2 , ηn2 ∼ σ 2 n, thus 1.6 holds. Therefore, the class of random variables in Theorems 1.1 is of very
broad range.
Remark 1.4. Whether Theorem 1.1 holds for 1/2 ≤ α < 1 remains open.
2. Proofs
Furthermore, the following four lemmas will be useful in the proof, and the first is due to
Csörgo et al. 15.
Lemma 2.1. Let X be a random variable with EX μ, and denote lx EX − μ2 I{|X − μ| ≤ x}.
The following statements are equivalent:
i lx is a slowly varying function at ∞;
ii X is in the domain of attraction of the normal law;
iii x2 P|X − μ| > x olx;
iv xE|X − μ|I|X − μ| > x olx;
v E|X − μ|α I|X − μ| ≤ x oxα−2 lx for α > 2.
4
Journal of Applied Mathematics
Lemma 2.2. Let {ξ, ξn }n∈N be a sequence of uniformly bounded random variables. If there exist
constants c > 0 and δ > 0 such that
δ
Eξk ξj ≤ c k ,
j
for 1 ≤ k < j,
2.1
then
n
1 dk ξk 0
n → ∞ Dn
k1
lim
2.2
a.s.,
where dk and Dn are defined by 1.7.
Proof. We can easily apply the similar arguments of 2.1 in Wu 16 to get Lemma 2.2, and
we omit the details here.
The following Lemma 2.3 can be directly verified.
Lemma 2.3. (i) cn,i 2bn,i − dn,i ;
ii
iii
iv
n
2
bn,i
2n − bn,1 ∼ 2n;
i1
n
n
n
n
−
∼ ;
3 6n 1 3
10n
10n
10n
− 4bn,1 ∼
.
3
3n 1
3
2
dn,i
i1
n
2
cn,i
i1
For every 1 ≤ i ≤ k ≤ n, let
X ki Xi − μ I Xi − μ ≤ ηk ,
Sk,i i
ck,j X kj ,
j1
2
Vk k
2
X kj .
2.3
j1
Lemma 2.4. Suppose that the assumptions of Theorem 1.1 hold. Then
⎧
⎫
⎪
⎪
n
⎨
⎬
Sk,k − ESk,k
1
Φx a.s. for any x ∈ R,
lim
dk I ≤
x
n → ∞ Dn
⎪
⎪
⎩ 10kl ηk /3
⎭
k1
k k n
1 lim
X i − μ > ηk
− EI
X i − μ > ηk
0 a.s.,
dk I
n → ∞ Dn
i1
i1
k1
⎞
⎛
⎞⎞
⎛ ⎛
2
2
n
V
V
1 lim
dk ⎝f ⎝ k ⎠ − Ef ⎝ k ⎠⎠ 0 a.s.,
n → ∞ Dn
kl ηk
kl ηk
k1
where dk and Dn are defined by 1.7 and f is a nonnegative, bounded Lipschitz function.
2.4
2.5
2.6
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5
Proof. By EX − μ 0, Lemma 2.1 iv, we have
o l ηn
.
EX ni ≤ E X − μ I X − μ > ηn ηn
2.7
Thus, by 1.5 and Lemma 2.3 iv,
2
Var X ni EX ni − EX ni
!2
∼ l ηn ,
! 10n l ηn : Bn2 .
Var cn,i X ni ∼
3
i1
n
2.8
By 1.5 and Lemma 2.3 i,
max cn,i ≤ 2max bn,i ≤ 2bn,1 ∼ 2 ln n,
1≤i≤n
ln n max1≤i≤n EX ni Bn
1≤i≤n
−→ 0.
2.9
Thus by combining Lemma 2.3 iv, 1.6, and 2.8, Lindeberg condition
n
n
!2
2
1 c 2
E cn,i X ni I|cn,i Xni −Ecn,i Xni |>εBn ∼ cn,i
EX ni I|Xni −EXni |>εBn /2 ln n
2
nl ηn i1
Bn i1
≤
n
2
c 2
EX ni I|Xni |>εBn /4 ln n
cn,i
nl ηn i1
n
2
c 2
cn,i
E X − μ Icηn / ln n<|X−μ|≤ηn nl ηn i1
l ηn − l cηn / ln n
c
−→ 0, as n −→ ∞
l ηn
2.10
hold.
Hence, it follows that
Sn,n − ESn,n d
−→ N,
Bn
as n −→ ∞.
2.11
This implies that for any gx, which is a nonnegative, bounded Lipschitz function,
Eg
Sn,n − ESn,n
Bn
−→ EgN,
as n −→ ∞.
2.12
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Journal of Applied Mathematics
Hence, we obtain
n
1 lim
dk Eg
n → ∞ Dn
k1
Sk,k − ESk,k
Bk
EgN
2.13
from the Toeplitz lemma.
On the other hand, note that 2.4 is equivalent to
n
1 dk g
lim
n → ∞ Dn
k1
Sk,k − ESk,k
Bk
EgN
2.14
a.s.,
from Theorem 7.1 of Billingsley 17 and Section 2 of Peligrad and Shao 18. Hence, to prove
2.4, it suffices to prove
n
Sk,k − ESk,k
Sk,k − ESk,k
1 lim
− Eg
0
dk g
n → ∞ Dn
Bk
Bk
k1
a.s.,
2.15
for any gx which is a nonnegative, bounded Lipschitz function.
Let
ξk g
Sk,k − ESk,k
Bk
− Eg
Sk,k − ESk,k
Bk
,
for k ≥ 1.
2.16
Clearly, there is a constant c > 0 such that
gx ≤ c,
gx − g y ≤ cx − y
for any x, y ∈ R,
|ξk | ≤ 2c, for any k.
2.17
For any 1 ≤ k < l, note that
Sl,l − Sl,k l
i1
cl,i X li −
k
l
cl,i X li cl,i X li .
i1
ik1
2.18
Journal of Applied Mathematics
7
For any 1 ≤ k < j, note that gSk,k −ESk,k /Bk and gSj,j −ESj,j −Sj,k −ESj,k /Bj are
2
2
2
∼ 10k/3, cj,i
≤ 4bj,i
,
independent, gx is a nonnegative, bounded Lipschitz function, ki1 ck,i
k 2
1/4
, x ≥ 1. By the definition of ηj , we get
i1 bk,i ∼ 2k, and ln x ≤ 4x
Sj,j − ESj,j
S
−
ES
k,k
k,k
Eξk ξj Cov g
,g
Bk
Bj
⎛
⎛
! ⎞⎞
S
−
ES
−
S
−
ES
j,j
j,j
j,k
j,k
S
−
ES
Sk,k − ESk,k
j,j
j,j
⎜
⎜
⎟⎟
Cov⎝g
,g
− g⎝
⎠⎠
Bk
Bj
Bj
!
E ki1 cj,i X ji − EX ji ≤c
≤c
jl ηj
"
≤c
k
i1
jl ηj
c
≤c
k
i1
2
2
bj,i
EX ji
≤c
k
i1
"
E
k
i1 cj,i X ji − EX ji jl ηj
!2
2 bk,i bj,k1 l ηj
jl ηj
2
2
bk,i
ki1 bj,k1
k k ln2 j/k
≤c
j
j
1/4
k
.
j
2.19
By Lemma 2.2, 2.15 holds.
Now we prove 2.5. Let
k k X i − μ > ηk
− EI
X i − μ > ηk
Zk I
i1
for any k ≥ 1.
2.20
i1
It is known that IA ∪ B − IB ≤ IA for any sets A and B; then for 1 ≤ k < j, by Lemma 2.1
iii and 1.5, we get
l ηj
o1
P X − μ > ηj o1 2 .
j
ηj
2.21
8
Journal of Applied Mathematics
Hence for 1 ≤ k < j,
j
k EZk Zj Cov I
Xi − μ > ηk , I
Xi − μ > ηj
i1
i1
j
k Xi − μ > ηk , I
Xi − μ > ηj
Cov I
i1
i1
−I
j
Xi − μ > ηj
ik1
j
j
≤ EI
X i − μ > ηj
−I
X i − μ > ηj i1
ik1
2.22
k ≤ EI
X i − μ > ηj
≤ kP X − μ > ηj
i1
≤
k
.
j
By Lemma 2.2, 2.5 holds.
Finally, we prove 2.6. Let
⎞
⎛
⎞
2
2
V
V
ζk f ⎝ k ⎠ − Ef ⎝ k ⎠
kl ηk
kl ηk
⎛
for any k ≥ 1.
2.23
For 1 ≤ k < j,
⎛ ⎛
⎞ ⎛
2 ⎞⎞
2
V
V
j
Eζk ζj Cov⎝f ⎝ k ⎠, f ⎝ ⎠⎠
kl ηk
jl ηj
⎛ ⎛
⎛ 2 ⎞ ⎛
2 ⎞⎞
2 ⎞
2
k
Xi − μ ≤ ηj
V
V
−
−
μ
I
X
i
V
j
j
i1
⎠⎠
Cov⎝f ⎝ k ⎠, f ⎝ ⎠ − f ⎝
kl ηk
jl ηj
jl ηj
≤c
E
2 !
2 Xi − μ I Xi − μ ≤ ηj
kE X − μ I X − μ ≤ ηj
kl ηj
c
c jl ηj
jl ηj
jl ηj
k i1
k
c .
j
2.24
By Lemma 2.2, 2.6 holds. This completes the proof of Lemma 2.4.
Journal of Applied Mathematics
9
Proof of Theorem 1.1. Let Zj Tj /jj 1μ/2; then 1.8 is equivalent to
⎛ √
⎞
k
n
3μ
1 dk I ⎝ √
ln Zj ≤ x⎠ Φx,
lim
n → ∞ Dn
10Vk j1
k1
2.25
a.s. for any x,
where Φx is the distribution function of the standard normal random variable N.
Let q ∈ 4/3, 2, then E|X|q < ∞. Using Marcinkiewicz-Zygmund strong large number
law, we have
Sk − μk o k1/q
!
2.26
a.s.
Thus,
|Zi − 1| i
j1 Sj − ii 1μ/2
ii 1μ/2
i1/q−1 −→ 0,
≤
i j1 Sj − μj
ii 1μ/2
i
≤
j1
j 1/q
ii 1μ/2
≤c
i1/q1
i2
2.27
a.s.
Hence by | ln1 x − x| Ox2 for |x| < 1/2, for any 0 < ε < 1,
k
k
k
k
1
1
1
c
ln Zi − √
Zi − 1 ≤ c Zi − 12 ≤ i21/q−1
√
1 ± εBk i1
1 ± εBk i1
kl ηk i1
kl ηk i1
≤c
1
−→ 0
k3/2−2/q l ηk
a.s. k −→ ∞,
2.28
from 3/2 − 2/q > 0, lx is a slowly varying function at ∞, and ηk ≤ k 1.
Hence for almost every event ω and any δ > 0, there exists k0 k0 ω, δ, x such that
for k > k0 ,
I
√
μ
1 ± εBk
k
Zi − 1 ≤ x − δ
≤I
i1
√
≤I
√
μ
1 ± εBk
μ
1 ± εBk
k
i1
k
i1
ln Zi ≤ x
Zi − 1 ≤ x δ .
2.29
10
Journal of Applied Mathematics
Note that under condition |Xj − μ| ≤ ηk , 1 ≤ j ≤ k,
μ
k
i1
Zi − 1 k
i
i1
k
i l
Sl − μ il1 l 1
Xj − μ
ii 1/2
ii 1/2 l1 j1
i1
l1
i
i 2 i1−j
i k 1
Xj − μ Xj − μ
ii 1/2 j1 lj
ii 1
i1
i1 j1
k
k 2 i1−j
k k
X kj ck,j X kj
ii 1
j1 ij
j1
Sk,k .
Thus, for any given 0 < ε < 1, δ > 0, combining 2.29, we have for k > k0
√
I
⎛
⎞
√ k
k
!
3μ i1 ln Zi
3μ i1 ln Zi
2
⎜
⎟
≤ x ≤ I⎝ V
>
εkl
ηk
≤
x
I
1
√
⎠
k
10Vk
101 εkl ηk
k X i − μ > ηk
I
i1
k
!
μ i1 Zi − 1
2
≤I
≤ x δ I V k > 1 εkl ηk
√
1 εBk
k Xi − μ > ηk
I
i1
!
2
≤ x δ I V k > 1 εkl ηk
√
1 εBk
k Xi − μ > ηk
2I
for x ≥ 0,
≤I
i1
√
!
3μ ki1 ln Zi
2
Sk,k
≤x ≤I √
≤ x δ I V k < 1 − εkl ηk
√
1 − εBk
10Vk
k 2I
X i − μ > ηk
for x < 0,
√
!
3μ ki1 ln Zi
2
Sk,k
≤x ≥I √
≤ x − δ − I V k < 1 − εkl ηk
√
1 − εBk
10Vk
k − 2I
Xi − μ > ηk , for x ≥ 0,
I
I
Sk,k
i1
i1
2.30
Journal of Applied Mathematics
√ k
!
3μ i1 ln Zi
2
Sk,k
≤x ≥I √
≤ x − δ − I V k > 1 εkl ηk
I
√
10Vk
1 εBk
k Xi − μ > ηk , for x < 0.
− 2I
11
i1
2.31
Therefore, to prove 2.25, it suffices to prove
n
1 dk I
lim
n → ∞ Dn
k1
Sk,k √
≤ 1 ± εx ± δ1
Bk
Φ
√
1 ± εx ± δ1
k n
1 X i − μ > ηk
0
lim
dk I
n → ∞ Dn
i1
k1
!
a.s.,
2.32
a.s.,
2.33
n
!
2
1 dk I V k > 1 εkl ηk 0 a.s.,
n → ∞ Dn
k1
2.34
n
!
2
1 dk I V k < 1 − εkl ηk 0
n → ∞ Dn
k1
2.35
lim
lim
a.s,
for any 0 < ε < 1 and δ1 > 0.
Firstly, we prove 2.32. Let 0 < β < 1/2 and h· be a real function, such that for any
given x ∈ R,
I y≤
!
!
√
√
1 ± εx ± δ1 − β ≤ h y ≤ I y ≤ 1 ± εx ± δ1 β .
2.36
By EXi − μ 0, Lemma 2.1iv and 1.5, we have
k
k
ESk,k E ck,i Xi − μ I Xi − μ ≤ ηk ≤ 2 bk,i EXi − μI Xi − μ > ηk
i1
i1
j
k
k k o l ηk
1 1
E X − μI X − μ > ηk 2
j
j
ηk
i1 ji
j1 i1
o
kl ηk .
This, combining with 2.4, 2.36 and the arbitrariness of β in 2.36, 2.32 holds.
2.37
12
Journal of Applied Mathematics
By 2.5, 2.21 and the Toeplitz lemma,
n
n
k k 1 1 X i − μ > ηk
X i − μ > ηk
0≤
dk I
dk EI
∼
Dn k1
Dn k1
i1
i1
n
1 ≤
dk kP X − μ > ηk −→ 0 a.s.
Dn k1
2.38
That is, 2.33 holds.
Now we prove 2.34. For any given ε > 0, let f be a nonnegative, bounded Lipschitz
function such that
Ix > 1 ε ≤ fx ≤ I x > 1 2
2
ε!
.
2
2.39
2
From EV k klηk , X ki is i.i.d., EX ki − EX ki 0, Lemma 2.1 v, and 1.5,
2
P Vk > 1 2
2
ε ! !
ε !
kl ηk P V k − EV k > kl ηk
2
2
2
2 !2
E V k − EV k
≤c
k 2 l 2 ηk
4 EX − μ I X − μ ≤ ηk
≤c
kl2 ηk
2.40
o1ηk2
o1 −→ 0.
kl ηk
Therefore, combining 2.6 and the Toeplitz lemma,
⎛
⎞
2
n
n
!
2
V
1 1 dk I V k > 1 εkl ηk ≤
dk f ⎝ k ⎠
0≤
Dn k1
Dn k1
kl ηk
⎞
⎛
2
n
n
2
V
1 1 ε ! !
∼
kl ηk
dk Ef ⎝ k ⎠ ≤
dk EI V k > 1 Dn k1
Dn k1
2
kl ηk
2.41
n
2
1 ε ! !
kl ηk
dk P V k > 1 Dn k1
2
−→ 0 a.s.
Hence, 2.34 holds. By similar methods used to prove 2.34, we can prove 2.35. This
completes the proof of Theorem 1.1.
Journal of Applied Mathematics
13
Acknowledgments
The author is very grateful to the referees and the Editors for their valuable comments and
some helpful suggestions that improved the clarity and readability of the paper. This paper
is supported by the National Natural Science Foundation of China 11061012, a project supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands
in Guangxi Institutions of Higher Learning 2011 47, and the support program of Key Laboratory of Spatial Information and Geomatics 1103108-08.
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