A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law Journal of Inequalities and Applications 2012,
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A note on the almost sure limit theorem for self-normalized partial sums of
random variables in the domain of attraction of the normal law
Journal of Inequalities and Applications 2012, 2012:17 doi:10.1186/1029-242X-2012-17
Qunying Wu (wqy666@glite.edu.cn)
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Trang 2A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of
the normal law
Qunying Wu1,2
1College of Science, Guilin University of Technology,
Guilin 541004, P R China
2Guangxi Key Laboratory of Spatial Information and Geomatics,
Guilin 541004, P.R China Email address: wqy666@glite.edu.cn
Abstract
Let X, X1, X2, be a sequence of independent and identically distributed random variables in the domain of attraction
of a normal distribution A universal result in almost sure limit theorem for the self-normalized partial sums S n /V nis
established, where S n=Pn
i=1 X i , V2=Pn
i=1 X2
i Mathematical Scientific Classification: 60F15
Keywords: domain of attraction of the normal law; self-normalized partial sums; almost sure central limit theorem
1 Introduction
Throughout this article, we assume {X, X n}n∈N is a sequence of independent and identically distributed (i.i.d.)
random variables with a non-degenerate distribution function F For each n ≥ 1, the symbol S n /V n denotes
self-normalized partial sums, where S n=Pn
i=1 X i , V2=Pn
i=1 X2
i We say that the random variable X belongs to the domain
of attraction of the normal law, if there exist constants a n > 0, b n ∈ R such that
S n − b n
a n
d
where N is the standard normal random variable We say that {X n}n∈Nsatisfies the central limit theorem (CLT)
Trang 3It is known that (1) holds if and only if
lim
x→∞
x2P(|X| > x)
In contrast to the well-known classical central limit theorem, Gine et al [1] obtained the following self-normalized
version of the central limit theorem: (S n − ES n )/V n
d
−→ N as n → ∞ if and only if (2) holds.
Brosamler [2] and Schatte [3] obtained the following almost sure central limit theorem (ASCLT): Let {X n}n∈Nbe i.i.d random variables with mean 0, variance σ2> 0 and partial sums S n Then
lim
n→∞
1
D n
n
X
k=1
d k I
(
S k
σ√k < x
)
= Φ(x) a.s for all x ∈ R, (3)
with d k = 1/k and D n =Pn k=1 d k , where I denotes an indicator function, and Φ(x) 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 Cs´aki [7], H¨ormann [8], Wu [9, 10], and Ye and Wu [11] Huang and Zhang [12] and Zhang and Yang [13] obtained ASCLT results for self-normalized version
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, p 35]) shows
that the large the weight sequence {d k ; k ≥ 1} in (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], Equation (3) fails for weight d k = 1 The optimal weight sequence remains unknown
The purpose of this article is to study and establish the ASCLT for self-normalized 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 d k = k−1exp(ln k)α), 0 ≤ α < 1/2
Our theorem is formulated in a more general setting
2
Trang 4Theorem 1.1 Let {X, X n}n∈N be a sequence of i.i.d random variables in the domain of attraction of the normal law with mean zero Suppose 0 ≤ α < 1/2 and set
d k= exp(ln
αk)
k , D n=
n
X
k=1
Then
lim
n→∞
1
D n
n
X
k=1
d k I
(
S k
V k ≤ x
)
= Φ(x) a.s for any x ∈ R. (5)
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 {d k}k∈N by any {d∗
k}k∈N such that 0 ≤
d∗
k ≤ d k ,P∞k=1 d∗
k = ∞.
Remark 1.3 Our results not only give substantial improvements for weight sequence in theorem 1.1 obtained by Huang [12] but also removed the condition nP(|X1| > ηn ) ≤ c(log n)ε 0, 0 < ε0< 1 in theorem 1.1 of [12].
Remark 1.4 If EX2 < ∞, then X is in the domain of attraction of the normal law Therefore, the class of random
variables in Theorems 1.1 is of very broad range.
Remark 1.5 Essentially, the open problem should be whether Theorem 1.1 holds for 1/2 ≤ α < 1 remains open.
2 Proofs
In the following, a n ∼ b ndenotes limn→∞ a n /b n = 1 The symbol c stands for a generic positive constant which
may differ from one place to another
Furthermore, the following three lemmas will be useful in the proof, and the first is due to [15]
Lemma 2.1 Let X be a random variable with EX = 0, and denote l(x) = EX2I{|X| ≤ x} The following statements are equivalent:
(i) X is in the domain of attraction of the normal law.
(ii) x2P(|X| > x) = o(l(x)).
(iii) xE(|X|I(|X| > x)) = o(l(x)).
(iv) E(|X|αI(|X| ≤ x)) = o(xα−2l(x)) for α > 2.
Lemma 2.2 Let {ξ, ξ n}n∈N be a sequence of uniformly bounded random variables If exist constants c > 0 and δ > 0 such that
|Eξkξj | ≤ c k
j
!δ
, for 1 ≤ k < j, (6)
then
lim
n→∞
1
D n
n
X
k=1
d kξk= 0 a.s., (7)
where d k and D n are defined by (4).
3
Trang 5Proof Since
E
n
X
k=1
d kξk
2
≤
n
X
k=1
d2
kEξ2
1≤k< j≤n
d k d j|Eξkξj|
=
n
X
k=1
d2
kEξ2
1≤k< j≤n; j/k≥ln2/δD n
d k d j|Eξkξl| + 2 X
1≤k< j≤n; j/k<ln2/δD n
d k d j|Eξkξl|
By the assumption of Lemma 2.2, there exists a constant c > 0 such that |ξ k | ≤ c for any k Noting that exp(lnαx) =
exp(R1x α(ln u) uα−1du), we have exp(lnαx), α < 1 is a slowly varying function at infinity Hence,
T n1 ≤ c
n
X
k=1
exp(2 lnαk)
k2 ≤ c
∞
X
k=1
exp(2 lnαk)
k2 < ∞
By (6),
T n2 ≤ c X
1≤k< j≤n; j/k≥ln2/δD n
d k d j
k j
!δ
≤ c X
1≤k< j≤n; j/k≥ln2/δD n
d k d j
ln2D n
≤ cD
2
n
ln2D n
On the other hand, if α = 0, we have d k = e/k, D n ∼ e ln n, hence, for sufficiently large n,
T n3 ≤ c
n
X
k=1
1
k
k lnX2/δD n j=k
1
j ≤ cD n ln ln D n≤
D2
ln2D n
If α > 0, note that
D n ∼
Z n
1
exp(lnαx)
x dx =
Z ln n
0
exp(yα)dy
∼
Z ln n
0
exp(yα) +1 − α
α y
−αexp(yα)
!
dy
=
Z ln n
0
1 α
y1−αexp(yα)0dy
= 1
αln
1−αn exp(lnαn), n → ∞. (11)
This implies
ln D n∼ lnαn, exp(lnαn) ∼ αD n
(ln D n)1−αα
, ln ln D n ∼ α ln ln n.
4
Trang 6Thus combining |ξk | ≤ c for any k,
T n3 ≤ c
n
X
k=1
d k
X
1≤k< j≤n; j/k<(ln D n) 2/δ
d j
≤ c
n
X
k=1
d k
X
k< j≤k(ln D n) 2/δ
exp(lnαn)1
j
≤ c exp(lnαn) ln ln D n
n
X
k=1
d k
≤ c D
2ln ln D n
(ln D n)(1−α)/α Since α < 1/2 implies (1 − 2α)/(2α) > 0 and ε1:= 1/(2α) − 1 > 0 Thus, for sufficiently large n, we get
T n3 ≤ c D
2
n
(ln D n)1/(2α)
ln ln D n (ln D n)(1−2α)/(2α) ≤ D
2
n
(ln D n)1/(2α) = D
2
n
(ln D n)1+ε 1 (12)
Let T n:= 1
D n
Pn
k=1 d kξk, ε2:= min(1, ε1) Combining (8)-(12), for sufficiently large n, we get
ET2
(ln D n)1+ε 2
By (11), we have D n+1 ∼ D n Let 0 < η < ε2/(1 + ε2), n k = inf{n; D n ≥ exp(k1−η)}, then D n k ≥ exp(k1−η), D n k−1 <
exp(k1−η) Therefore
1 ≤ D n k
exp(k1−η) ∼
D n k−1
exp(k1−η) < 1 → 1, that is,
D n k ∼ exp(k1−η)
Since (1 − η)(1 + ε2) > 1 from the definition of η, thus for any ε > 0, we have
∞
X
k=1
P(|T n k | > ε) ≤ c
∞
X
k=1
ET n2k ≤ c
∞
X
k=1
1
k(1−η)(1+ε 2 ) < ∞
By the Borel-Cantelli lemma,
T n k → 0 a.s
Now for n k < n ≤ n k+1, by |ξk | ≤ c for any k,
|T n | ≤ |T n k| + c
D n k
n k+1
X
i=n k+1
d i ≤ |T n k | + c D n k+1
D n k
− 1
!
→ 0 a.s
5
Trang 7fromD nk+1
D nk ∼exp((k+1) exp(k1−η ) ) = exp(k1−η((1 + 1/k)1−η− 1)) ∼ exp((1 − η)k−η) → 1 I.e., (7) holds This completes the proof
of Lemma 2.2
Let l(x) = EX2I{|X| ≤ x}, b = inf{x ≥ 1; l(x) > 0} and
ηj= inf
(
s; s ≥ b + 1, l(s)
s2 ≤ 1
j
)
for j ≥ 1.
By the definition of ηj , we have jl(η j) ≤ η2
j and jl(η j− ε) > (ηj− ε)2for any ε > 0 It implies that
nl(η n) ∼ η2
For every 1 ≤ i ≤ n, let
¯
X ni = X i I(|X i| ≤ ηn ), ¯S n=
n
X
i=1
¯
X ni , ¯V n2=
n
X
i=1
¯
X ni2
Lemma 2.3 Suppose that the assumptions of Theorem 1.1 hold Then
lim
n→∞
1
D n
n
X
k=1
d k I
¯S k − E ¯S k
p
kl(η k) ≤ x
= Φ(x) a.s for any x ∈ R, (14)
lim
n→∞
1
D n
n
X
k=1
d k
I
k
[
i=1
(|X i| > ηk)
− EI
k
[
i=1
(|X i| > ηk)
= 0 a.s., (15)
lim
n→∞
1
D n
n
X
k=1
d k
f
¯V2
k
kl(η k)
− E f
¯V2
k
kl(η k)
= 0 a.s., (16)
where d k and D n are defined by (4) and f is a non-negative, bounded Lipschitz function.
Proof By the cental limit theorem for i.i.d random variables and Var ¯S n ∼ nl(η n ) as n → ∞ from EX = 0,
Lemma 2.1 (iii), and (13), it follows that
¯S n − E ¯S n
p
nl(η n)
d
−→ N, as n → ∞,
where N denotes the standard normal random variable This implies that for any g(x) which is a non-negative,
bounded Lipschitz function
6
Trang 8
¯Spn − E ¯S n
nl(η n)
−→ Eg(N), as n → ∞,
Hence, we obtain
lim
n→∞
1
D n
n
X
k=1
d k Eg
¯Spk − E ¯S k
kl(η k)
= Eg(N)
from the Toeplitz lemma
On the other hand, note that (14) is equivalent to
lim
n→∞
1
D n
n
X
k=1
d k g
¯Spk − E ¯S k
kl(η k)
= Eg(N) a.s.
from Theorem 7.1 of [16] and Section 2 of [17] Hence, to prove (14), it suffices to prove
lim
n→∞
1
D n
n
X
k=1
d k
g
¯Spk − E ¯S k
kl(η k)
− Eg
¯Spk − E ¯S k
kl(η k)
= 0 a.s., (17)
for any g(x) which is a non-negative, bounded Lipschitz function.
For any k ≥ 1, let
ξk = g
¯Spk − E ¯S k
kl(η k)
− Eg
¯Spk − E ¯S k
kl(η k)
For any 1 ≤ k < j, note that g ¯S√k −E ¯S k
kl(η k)
!
and g
¯S j −E ¯S j−
k
P
i=1 (X i −EX i )I(|X i|≤ηj)
√
jl(η j)
are independent and g(x) is a non-negative,
bounded Lipschitz function By the definition of ηj, we get,
|Eξkξj| =
Cov
g
¯Spk − E ¯S k
kl(η k)
, g
¯Spj − E ¯S j
jl(η j)
=
Cov
g
¯Spk − E ¯S k
kl(η k)
, g
¯Spj − E ¯S j
jl(η j)
− g
¯S j − E ¯S j−Pk
i=1 (X i − EX i )I(|X i| ≤ ηj) p
jl(η j)
≤ c
E Pk
i=1 (X i − EX i )I(|X i| ≤ ηj)
!
p
jl(η j) ≤ c
p
kEX2I(|X| ≤ η j) p
jl(η j)
= c k
j
!1/2
7
Trang 9By Lemma 2.2, (17) holds.
Now we prove (15) Let
Z k = I
k
[
i=1
(|X i| > ηk)
− EI
k
[
i=1
(|X i| > ηk)
for any k ≥ 1.
It is known that I(A ∪ B) − I(B) ≤ I(A) for any sets A and B, then for 1 ≤ k < j, by Lemma 2.1 (ii) and (13), we
get
P(|X| > η j ) = o(1) l(η j)
η2
j
=o(1)
Hence
|EZ k Z j| =
Cov
I
k
[
i=1
(|X i| > ηk)
, I
j
[
i=1
(|X i| > ηj)
=
Cov
I
k
[
i=1
(|X i| > ηk)
, I
j
[
i=1
(|X i| > ηj)
− I
j
[
i=k+1
(|X i| > ηj)
≤ E
I
j
[
i=1
(|X i| > ηj)
− I
j
[
i=k+1
(|X i| > ηj)
≤ EI
k
[
i=1
(|X i| > ηj)
≤ kP(|X| > η j)
≤k
j.
By Lemma 2.2, (15) holds
Finally, we prove (16) Let
ζk = f
¯V2
k
kl(η k)
− E f
¯V2
k
kl(η k)
for any k ≥ 1.
For 1 ≤ k < j,
|Eζkζj| =
Cov
f
¯V2
k
kl(η k)
, f
¯V
2
j
jl(η j)
8
Trang 10Cov
f
¯V2
k
kl(η k)
, f
¯V
2
j
jl(η j)
− f
¯V2
j −Pk
i=1 X2
i I(|X i| ≤ ηj)
jl(η j)
≤ c
E Pk
i=1 X2
i I(|X i| ≤ ηj)
!
jl(η j) = c
kEX2I(|X| ≤ η j)
jl(η j) = c
kl(η j)
jl(η j)
= c k
j.
By Lemma 2.2, (16) holds This completes the proof of Lemma 2.3
Proof of Theorem 1.1 For any given 0 < ε < 1, note that
I S k
V k ≤ x
!
≤ I
p ¯S k
(1 + ε)kl(η k) ≤ x
+ I¯V k2> (1 + ε)kl(η k)+ I
k
[
i=1
|X i| > ηk)
, for x ≥ 0,
I S k
V k
≤ x
!
≤ I
p ¯S k
(1 − ε)kl(η k) ≤ x
+ I¯V k2< (1 − ε)kl(η k)+ I
k
[
i=1
|X i| > ηk)
, for x < 0,
and
I S k
V k ≤ x
!
≥ I
p ¯S k
(1 − ε)kl(η k) ≤ x
− I¯V k2< (1 − ε)kl(η k)− I
k
[
i=1
|X i| > ηk)
, for x ≥ 0,
I S k
V k ≤ x
!
≥ I
p ¯S k
(1 + ε)kl(η k) ≤ x
− I¯V2
k > (1 + ε)kl(η k)− I
k
[
i=1
|X i| > ηk)
, for x < 0.
Hence, to prove (5), it suffices to prove
lim
n→∞
1
D n
n
X
k=1
d k I
p¯S k
kl(η k)≤
√
1 ± εx
= Φ(√1 ± εx) a.s., (19)
lim
n→∞
1
D n
n
X
k=1
d k I
k
[
i=1
|X i| > ηk)
= 0 a.s., (20)
lim
n→∞
1
D n
n
X
k=1
d k I( ¯V2
k > (1 + ε)kl(η k)) = 0 a.s., (21)
9
Trang 11n→∞
1
D n
n
X
k=1
d k I( ¯V2
k < (1 − ε)kl(η k)) = 0 a.s (22)
by the arbitrariness of ε > 0
Firstly, we prove (19) Let 0 < β < 1/2 and h(·) be a real function, such that for any given x ∈ R,
I(y ≤ √1 ± εx − β) ≤ h(y) ≤ I(y ≤ √1 ± εx + β). (23)
By EX = 0, Lemma 2.1 (iii) and (13), we have
|E ¯S k | = |kEXI(|X| ≤ η k )| = |kEXI(|X| > η k )| ≤ kE|X|I(|X| > η k ) = o(pkl(η k))
This, combining with (14), (23) and the arbitrariness of β in (23), (19) holds
By (15), (18) and the Toeplitz lemma,
0 ≤ 1
D n
n
X
k=1
d k I
k
[
i=1
|X i| > ηk)
∼ D1n
n
X
k=1
d k EI
k
[
i=1
|X i| > ηk)
≤ 1
D n
n
X
k=1
d k kP(|X| > η k) → 0 a.s
That is (20) holds
Now we prove (21) For any µ > 0, let f be a non-negative, bounded Lipschitz function such that
I(x > 1 + µ) ≤ f (x) ≤ I(x > 1 + µ/2).
Form E ¯V2
k = kl(η k), ¯X niis i.i.d., Lemma 2.1 (iv), and (13),
P
¯V2
k >
1 + µ 2
kl(η k)
= P
¯V2
k − E ¯V k2> µ
2kl(η k)
≤ c E( ¯V
2
k − E ¯V2
k)2
k2l2(ηk) ≤ c
EX4I(|X| ≤ η k)
kl2(ηk)
= o(1)η
2
k
kl(η k) = o(1) → 0.
10
... Trang 9By Lemma 2.2, (17) holds.
Now we prove (15) Let
Z k = I
...
It is known that I (A ∪ B) − I(B) ≤ I (A) for any sets A and B, then for ≤ k < j, by Lemma 2.1 (ii) and (13), we
get
P(|X| > η j ) = o(1) l(η... j − E ¯S j−Pk
i=1 (X i − EX i )I(|X i| ≤ ηj) p