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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 ([email protected])
ISSN 1029-242X
Article type Research
Submission date 4 August 2011
Acceptance date 20 January 2012
Publication date 20 January 2012
Article URL http://www.journalofinequalitiesandapplications.com/content/2012/1/17
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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
Qunying Wu
1,2
1
College of Science, Guilin University of Technology,
Guilin 541004, P. R. China

X
2
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
=

n
i=1
X
i
, V
2
n
=


EX
2
I(|X| ≤ x)
= 0. (2)
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∈N
be
i.i.d. random variables with mean 0, variance σ
2
> 0 and partial sums S
n
. Then
lim
n→∞
1
D
n
n


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
−1
exp(ln k)
α
), 0 ≤ α < 1/2.
Our theorem is formulated in a more general setting.
2
Theorem 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

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
,

∞
k=1
d
∗
k

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) = EX
2
I{|X| ≤ x}. The following statements
are equivalent:
(i) X is in the domain of attraction of the normal law.
(ii) x
2
P(|X| > x) = o(l(x)).
(iii) xE(|X|I(|X| > x)) = o(l(x)).
(iv) E(|X|
α
I(|X| ≤ x)) = o(x
α−2
l(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
|Eξ
k
ξ
j
| ≤ c

k
j



n

k=1
d
k
ξ
k







2
≤
n

k=1
d
2
k
Eξ
2
k
+ 2

1≤k< j≤n
d

ξ
l
| + 2

1≤k< j≤n; j/k<ln
2/δ
D
n
d
k
d
j
|Eξ
k
ξ
l
|
:= T
n1
+ 2(T
n2
+ T
n3
). (8)
By the assumption of Lemma 2.2, there exists a constant c > 0 such that |ξ
k
| ≤ c for any k. Noting that exp(ln
α
x) =
exp(

By (6),
T
n2
≤ c

1≤k< j≤n; j/k≥ln
2/δ
D
n
d
k
d
j

k
j

δ
≤ c

1≤k< j≤n; j/k≥ln
2/δ
D
n
d
k
d
j
ln
2

1
j
≤ cD
n
ln ln D
n
≤
D
2
n
ln
2
D
n
. (10)
If α > 0, note that
D
n
∼

n
1
exp(ln
α
x)
x
dx =

ln n
0

α
)


dy
=
1
α
ln
1−α
n exp(ln
α
n), n → ∞. (11)
This implies
ln D
n
∼ ln
α
n, exp(ln
α
n) ∼
αD
n
(ln D
n
)
1−α
α
, ln ln D
n

2/δ
exp(ln
α
n)
1
j
≤ c exp(ln
α
n) ln ln D
n
n

k=1
d
k
≤ c
D
2
n
ln ln D
n
(ln D
n
)
(1−α)/α
.
Since α < 1/2 implies (1 − 2α)/(2α) > 0 and ε
1
:= 1/(2α) − 1 > 0. Thus, for sufficiently large n, we get
T

1+ε
1
. (12)
Let T
n
:=
1
D
n

n
k=1
d
k
ξ
k
, ε
2
:= min(1, ε
1
). Combining (8)-(12), for sufficiently large n, we get
ET
2
n
≤
c
(ln D
n
)
1+ε

D
n
k
exp(k
1−η
)
∼
D
n
k
−1
exp(k
1−η
)
< 1 → 1,
that is,
D
n
k
∼ exp(k
1−η
).
Since (1 − η)(1 + ε
2
) > 1 from the definition of η, thus for any ε > 0, we have
∞

k=1
P(|T
n

| ≤ c for any k,
|T
n
| ≤ |T
n
k
| +
c
D
n
k
n
k+1

i=n
k
+1
d
i
≤ |T
n
k
| + c

D
n
k+1
D
n
k

j
= inf

s; s ≥ b + 1,
l(s)
s
2
≤
1
j

for j ≥ 1.
By the definition of η
j
, we have jl(η
j
) ≤ η
2
j
and jl(η
j
− ε) > (η
j
− ε)
2
for any ε > 0. It implies that
nl(η
n
) ∼ η
2


i=1
¯
X
2
ni
.
Lemma 2.3. Suppose that the assumptions of Theorem 1.1 hold. Then
lim
n→∞
1
D
n
n

k=1
d
k
I







¯
S
k
− E





I








k

i=1
(|X
i
| > η
k
)








− EI



= 0 a.s., (15)
lim
n→∞
1
D
n
n

k=1
d
k






f






¯
V
2
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

nl(η
n
)


−→ Eg(N), as n → ∞,
Hence, we obtain
lim
n→∞
1
D
n
n

k=1
d
k
Eg







¯
S
k
− E
¯
S
k

kl(η

¯
S
k
− 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

k=1
d
k




− Eg







¯
S
k
− E
¯
S
k

kl(η
k
)
















− Eg







¯
S
k
− E
¯
S
k

kl(η
k
)





j
−E
¯
S
j
−
k

i=1
(X
i
−EX
i
)I(|X
i
|≤η
j
)
√
jl(η
j
)












¯
S
k
− E
¯
S
k

kl(η
k
)







, g







¯


=












Cov





















¯
S
j
− E
¯
S
j

jl(η
j
)







− g








jl(η
j
)



























| ≤ η
j
)


jl(η
j
)
≤ c

kEX
2
I(|X| ≤ η
j
)

jl(η
j
)
= c

k
j

1/2
.
7
By Lemma 2.2, (17) holds.
Now we prove (15). Let
Z






k

i=1
(|X
i
| > η
k
)








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
)









k

i=1
(|X
i
| > η
k
)








, I










=







Cov








I









| > η
j
)








− I








j

i=k+1
(|X
i
| > η
j
)











j

i=1
(|X
i
| > η
j
)








− I











k

i=1
(|X
i
| > η
j
)








≤ kP(|X| > η
j
)
≤
k
j
.
By Lemma 2.2, (15) holds.
Finally, we prove (16). Let

V
2
k
kl(η
k
)






for any k ≥ 1.
For 1 ≤ k < j,
|Eζ
k
ζ
j
| =







Cov






¯
V
2
j
jl(η
j
)























f






¯
V
2
k
kl(η
k
)






, f







¯

¯
V
2
j
−
k

i=1
X
2
i
I(|X
i
| ≤ η
j
)
jl(η
j
)















≤ c
E

k

i=1
X
2
i
I(|X
i
| ≤ η
j
)

jl(η
j
)
= c
kEX
2
I(|X| ≤ η
j
)
jl(η
j
)

S
k

(1 + ε)kl(η
k
)
≤ x







+ I

¯
V
2
k
> (1 + ε)kl(η
k
)

+ I












¯
S
k

(1 − ε)kl(η
k
)
≤ x







+ I

¯
V
2
k
< (1 − ε)kl(η
k
)

S
k
V
k
≤ x

≥ I







¯
S
k

(1 − ε)kl(η
k
)
≤ x







− I





, for x ≥ 0,
I

S
k
V
k
≤ x

≥ I







¯
S
k

(1 + ε)kl(η
k
)
≤ x


| > η
k
)








, for x < 0.
Hence, to prove (5), it suffices to prove
lim
n→∞
1
D
n
n

k=1
d
k
I







k
I








k

i=1
|X
i
| > η
k
)








= 0 a.s., (20)
lim
n→∞
1

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(

kl(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

k=1
d
k
EI








k

i=1
|X
i
| > η
k
)


¯
X
ni
is i.i.d., Lemma 2.1 (iv), and (13),
P

¯
V
2
k
>

1 +
µ
2

kl(η
k
)

= P

¯
V
2
k
− E
¯
V
2

4
I(|X| ≤ η
k
)
kl
2
(η
k
)
=
o(1)η
2
k
kl(η
k
)
= o(1) → 0.
10
Therefore, from (16) and the Toeplitz lemma,
0 ≤
1
D
n
n

k=1
d
k
I


)






∼
1
D
n
n

k=1
d
k
E f






¯
V
2
k
kl(η
k
)


k=1
d
k
P(
¯
V
2
k
> (1 + µ/2)kl(η
k
))
→ 0 a.s.
Hence, (21) holds. By similar methods used to prove (21), we can prove (22). This completes the proof of Theo-
rem 1.1.
Competing interests
The author declares that she has no competing interests.
Acknowledgments
The author was very grateful to the referees and the Editors for their valuable comments and some helpful sug-
gestions that improved the clarity and readability of the paper. This work was supported by the National Natural
Science Foundation of China (11061012), the 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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