Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 656257, 9 pages
http://dx.doi.org/10.1155/2013/656257
Research Article
The Almost Sure Local Central Limit Theorem for
the Negatively Associated Sequences
Yuanying Jiang1,2 and Qunying Wu1
1
2
College of Science, Guilin University of Technology, Guilin 541004, China
School of Statistics, Renmin University of China, Beijing 100872, China
Correspondence should be addressed to Yuanying Jiang; jyy@ruc.edu.cn
Received 13 May 2013; Accepted 18 June 2013
Academic Editor: Ying Hu
Copyright © 2013 Y. Jiang and Q. 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.
In this paper, the almost sure central limit theorem is established for sequences of negatively associated random variables:
𝑛
lim𝑛 → ∞ (1/ log 𝑛) ∑𝑘=1 (𝐼(𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 )/𝑘)𝑃(𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 ) = 1, almost surely. This is the local almost sure central limit theorem for
negatively associated sequences similar to results by Csáki et al. (1993). The results extend those on almost sure local central limit
theorems from the i.i.d. case to the stationary negatively associated sequences.
1. Introduction
Definition 1. Random variables 𝑋1 , 𝑋2 , . . . , 𝑋𝑛 , 𝑛 ≥ 2 are
said to be negatively associated (NA) if for every pair of
disjoint subsets 𝐴 1 and 𝐴 2 of {1, 2, . . . , 𝑛},
[2] for the moment inequalities, and Wu and Jiang [3] for
Chover’s law of the iterated logarithm.
Assume that {𝑋𝑛 , 𝑛 ≥ 1} is a strictly stationary sequence
of NA random variables with 𝐸𝑋1 = 0, 0 < 𝐸𝑋12 < ∞. Define
𝑆𝑛 = ∑𝑛𝑗=1 𝑋𝑗 ,
Cov (𝑓1 (𝑋𝑖 , 𝑖 ∈ 𝐴 1 ) , 𝑓2 (𝑋𝑗 , 𝑗 ∈ 𝐴 2 )) ≤ 0,
𝜎𝑛2 := Var 𝑆𝑛 ,
(1)
(2)
∞
where 𝑓1 and 𝑓2 are increasing for every variable (or decreasing for every variable) such that this covariance exists. A
sequence of random variables {𝑋𝑖 , 𝑖 ≥ 1} is said to be NA
if every finite subfamily of {𝑋𝑖 , 𝑖 ≥ 1} is NA.
Obviously, if {𝑋𝑖 , 𝑖 ≥ 1} is a sequence of NA random
variables and {𝑓𝑖 , 𝑖 ≥ 1} is a sequence of nondecreasing
(or nonincreasing) functions, then {𝑓𝑖 (𝑋𝑖 ), 𝑖 ≥ 1} is also a
sequence of NA random variables.
This definition was introduced by the Joag-Dev and
Proschan [1]. Statistical test depends greatly on sampling,
and the random sampling without replacement from a finite
population is NA, but it is not independent. NA sampling
has wide applications such as those in multivariate statistical
analysis and reliability theory. Because of the wide applications of NA sampling, the notions of NA random variables
have received more and more attention recently. We refer to
Joag-Dev and Proschan [1] for fundamental properties, Shao
𝜎2 := Var 𝑋1 + 2 ∑Cov (𝑋1 , 𝑋𝑗 ) .
𝑗=2
(3)
(1) Newman [4] and Matuła [5] showed that NA stationary sequences satisfy the central limit theorem (CLT)
under 𝜎2 > 0, that is,
󵄨󵄨
󵄨󵄨
𝑆
󵄨
󵄨
sup 󵄨󵄨󵄨𝑃 ( 𝑛 < 𝑥) − Φ (𝑥)󵄨󵄨󵄨 = 𝑜 (1) .
(4)
󵄨󵄨
𝜎𝑛
𝑥∈𝑅 󵄨󵄨
(2) Applying Matuła [6] and Wu’s [7] methods, we can
easily show that NA sequences satisfy the almost sure
central limit theorem (ASCLT), that is,
1 𝑛 1
∑ 𝐼 {𝑆𝑘 ≤ 𝑥𝜎𝑘1/2 } = Φ (𝑥)
𝑛 → ∞ log 𝑛
𝑘
𝑘=1
lim
a.s. ∀𝑥 ∈ R, (5)
where Φ(𝑥) is the standard normal distribution function and 𝐼{𝐴} denotes the indicator of the event 𝐴.
2
Journal of Applied Mathematics
The ASCLT was stimulated by Brosamler [8] and Schatte
[9]. Both were concerned with the partial sum of independent
and identically distributed (i.i.d.) random variables with
more than the second moment. The ASCLT was extensively
studied in the past two decades and an interesting direction of
the study is to prove it for dependent variables. There are some
results for weakly dependent variables such as 𝛼, 𝜌, 𝜙-mixing
and associated random variables. Among those results, we
refer to Peligrad and Shao [10], Matuła [11], and Wu [7].
More general version of ASCLT was proved by Csáki et al.
[12]. The following theorem is due to them.
Theorem A. Let {𝑋𝑛 , 𝑛 ≥ 1} be a sequence of i.i.d. random
variables with 𝐸|𝑋1 |3 < ∞, let 𝐸𝑋1 = 0. 𝑎𝑘 , 𝑏𝑘 satisfy
−∞ ≤ 𝑎𝑘 ≤ 0 ≤ 𝑏𝑘 ≤ ∞,
𝑘 = 1, 2, . . . ,
(6)
log 𝑘
= 𝑂 (log 𝑛) ,
≤ 𝑆𝑘 < 𝑏𝑘 )
∑
𝑛
𝛼𝑘
𝑘
𝑘=1
𝜇𝑛 := ∑
(11)
under certain conditions.
In our considerations, we will need the following CoxGrimmett coefficient which describes the covariance structure of the sequence
𝑢 (𝑛) := sup ∑
𝑘∈𝑁𝑗:|𝑗−𝑘|≥𝑛
󵄨󵄨
󵄨
󵄨󵄨Cov (𝑋𝑗 , 𝑋𝑘 )󵄨󵄨󵄨 ,
󵄨
󵄨
𝑛 ∈ 𝑁 ∪ {0} .
3/2
𝑘=1 𝑘 𝑃 (𝑎𝑘
(12)
We remark that for a stationary sequence of NA random
variables
∞
𝑢 (𝑛) = −2 ∑ Cov (𝑋1 , 𝑋𝑘 ) ,
and assume that
𝑛
So we need to investigate the limit behavior of
𝑛 ∈ 𝑁.
(13)
𝑘=𝑛+1
𝑎𝑠 𝑛 󳨀→ ∞,
(7)
and then
1 𝑛 𝐼 {𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 }
∑
=1
𝑛 → ∞ log 𝑛
𝑘𝑃 (𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 )
𝑘=1
lim
a.s.
(8)
This result may be called almost sure local central limit
theorem, while (5) may be called almost sure global central
limit theorem. Hurelbaatar [13] extended (8) to the case of 𝜌mixing sequences and Weng et al. [14] derived an almost sure
local central limit theorem for the product of partial sums
of a sequence of i.i.d. positive random variables under some
regular conditions. For more details, we refer to Berkes and
Csáki [15] and Földes [16].
Our concern in this paper is to give a common generalization of (8) to the case of NA sequences.
In the next section we present the exact results, postponing some technical lemmas and the proofs to Section 3.
Assume in the following that {𝑋𝑛 , 𝑛 ≥ 1} is a strictly
stationary sequence of NA random variables with 𝐸𝑋1 =
0, 0 < 𝐸𝑋12 < ∞. We consider the limit behavior of the
logarithmic average
∞
∑ 𝑢 (𝑛) < ∞,
(14)
𝑛=1
and for some 𝛽 > 1,
1/3
(log 𝑘)
𝑘𝑝𝑘
1≤𝑘≤𝑛
∑
2
−𝛽
= 𝑂 ((log 𝑛) (log log 𝑛) ) .
(15)
Then we have
lim
𝑛→∞
𝜇𝑛
= 1,
log 𝑛
a.s.,
(16)
where 𝜇𝑛 is defined by (11).
(9)
with −∞ ≤ 𝑎𝑘 ≤ 0 ≤ 𝑏𝑘 ≤ ∞, where the terms in the sum
above are defined to be 1 if their denominator happens to be
0.
More precisely let {𝑎𝑛 , 𝑛 ≥ 1} and {𝑏𝑛 , 𝑛 ≥ 1} be two
sequences of real numbers and put
Remark 3. Let 𝑎𝑘 = −∞ and 𝑏𝑘 = 𝑥𝜎𝑘1/2 in (6). By the central
limit theorem (4), we have 𝑝𝑘 = 𝑃(𝑆𝑘 /𝜎𝑘1/2 < 𝑥) → Φ(𝑥),
obviously (15) satisfies; then (16) becomes (5), which is the
almost sure global central limit theorem. Thus the almost sure
local central limit theorem is a general result which contains
the almost sure global central limit theorem.
Remark 4. The condition (15) is satisfied with a wide range of
𝑝𝑘 ; for example, if
𝑝𝑘 := 𝑃 (𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 ) ,
𝐼 {𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 }
{
{
, if 𝑝𝑘 ≠ 0,
𝛼𝑘 := {
𝑝𝑘
{
if 𝑝𝑘 = 0.
{1,
Theorem 2. Let {𝑋𝑛 , 𝑛 ≥ 1} be a strictly stationary sequence
of NA random variables with 𝐸𝑋1 = 0, 𝐸|𝑋1 |3 < ∞ and let
𝜎2 > 0. 𝑎𝑘 , 𝑏𝑘 satisfy (6). Assume that
𝑝𝑘 ≠ 0
2. Main Results
1 𝑛 𝐼 {𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 }
∑
lim
𝑛 → ∞ log 𝑛
𝑘𝑃 (𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 )
𝑘=1
By Lemma 8 of Newman [4], we have 𝑢(0) < ∞ and
lim𝑛 → ∞ 𝑢(𝑛) = 0.
In the following, 𝜉𝑛 ∼ 𝜂𝑛 denotes 𝜉𝑛 /𝜂𝑛 → 1, 𝑛 → ∞.
𝜉𝑛 = 𝑂(𝜂𝑛 ) denotes that there exists a constant 𝑐 > 0 such
that 𝜉𝑛 ≤ 𝑐𝜂𝑛 for sufficiently large 𝑛. The symbols 𝑐, 𝑐1 , 𝑐2 , . . .,
stand for generic positive constants which may differ from
one place to another.
(10)
𝛽
𝑝𝑘 = 0 or 𝑝𝑘 ≥ 𝑐
(log log 𝑘)
2/3
(log 𝑘)
(17)
Journal of Applied Mathematics
3
holds, then the condition (15) is satisfied. In fact, letting 0 <
𝛿 < 1, we have
with some 𝛽 > 1 and positive constant 𝑐, then
1 𝑛 1
∑ 𝜉𝑘 = 1 a.s.
𝑛 → ∞ log 𝑛
𝑘
𝑘=1
1/3
lim
(log 𝑘)
𝑘𝑝𝑘
1≤𝑘≤𝑛
∑
𝑝𝑘 ≠ 0
(24)
The following Lemma 8 is obvious.
1−𝛿
𝛿 (log 𝑘)
−𝛽
≤ 𝑐 ∑ (log log 𝑘) (log 𝑘)
1≤𝑘≤𝑛
𝑘
(18)
1−𝛿
(log 𝑘)
≤ 𝑐(log log 𝑛) (log 𝑛) ∑
𝑘
1≤𝑘≤𝑛
−𝛽
𝛿
−𝛽
2
Lemma 8. Assume that the nonnegative random sequence
{𝜉𝑛 , 𝑛 ≥ 1} satisfies (24) and the sequence {𝜂𝑛 , 𝑛 ≥ 1} is such
that, for any 𝜀 > 0, there exists a 𝑘0 = 𝑘0 (𝜀, 𝜔) for which
(1 − 𝜀) 𝜉𝑘 ≤ 𝜂𝑘 ≤ (1 + 𝜀) 𝜉𝑘 ,
(25)
a.s.
(26)
Then we have also
≤ 𝑐(log log 𝑛) (log 𝑛)
2
𝑘 > 𝑘0 .
1 𝑛 1
∑ 𝜂𝑘 = 1
𝑛 → ∞ log 𝑛
𝑘
𝑘=1
lim
−𝛽
= 𝑂 ((log 𝑛) (log log 𝑛) ) .
In the given theorem below, we strengthen the condition
(6) on 𝑎𝑘 and 𝑏𝑘 . Meanwhile, as a compensation, we do not
need to impose restricting condition (15) on 𝑝𝑘 .
Theorem 5. Let {𝑋𝑛 , 𝑛 ≥ 1} be a strictly stationary sequence of
NA random variables with 𝐸𝑋1 = 0, 𝐸|𝑋1 |3 < ∞, and 𝜎2 > 0,
and let 𝑎𝑘 and 𝑏𝑘 satisfy
−𝑐1 𝑘1/2−𝛼 ≤ 𝑎𝑘 ≤ −𝑐2 𝑘1/2−𝛼 ,
The following Lemma 9 is an easy corollary to the Corollary 2.2 in Matuła [5] under strictly stationary condition,
which studies the rate of convergence in the CLT under negative dependence. It was also studied in Pan [17]. Of course
this is the Berry-Esseen inequality for the NA sequence.
Lemma 9. Let {𝑋𝑗 , 𝑗 ∈ 𝑁} be a strictly stationary sequence
of NA random variables with 𝐸𝑋1 = 0, 𝐸|𝑋1 |3 < ∞, 𝜎2 > 0
satisfying (14). Then one has
(19)
𝑐3 𝑘1/2−𝛼 ≤ 𝑏𝑘 ≤ 𝑐4 𝑘1/2−𝛼 ,
where 0 < 𝛼 < 1/7. Assume that (14) hold, and then we have
(16).
󵄨󵄨
󵄨󵄨
𝑆
󵄨
󵄨
sup 󵄨󵄨󵄨𝑃 ( 𝑛 < 𝑥) − Φ (𝑥)󵄨󵄨󵄨 = 𝑂 (𝑛−1/5 ) .
󵄨󵄨
󵄨
𝜎
𝑥∈𝑅 󵄨
𝑛
(27)
Lemma 10. If the conditions of Lemma 9 hold, 𝑎𝑘 and 𝑏𝑘 satisfy
(19). Then one has
𝑐1󸀠 𝑘−𝛼 ≤ 𝑝𝑘 ≤ 𝑐2󸀠 𝑘−𝛼
3. Proofs
(𝑘 ≥ 𝑘0 ) ,
(28)
The following lemmas play important roles in the proof of our
theorems. The proofs are given in the Appendix.
where 𝛼 is as (19).
Lemma 6. Assume that {𝜉𝑛 , 𝑛 ≥ 1} are random variables such
that
Lemma 11. If the conditions of Lemma 9 hold, 𝑎𝑘 and 𝑏𝑘 satisfy
(19), and 𝛼 is as (19). Assume that 𝜀𝑙 = 𝑙3𝛼/2 , and then the
following asymptotic relations hold:
𝜉𝑘 ≥ 0,
𝐸𝜉𝑘 = 1,
𝑘 = 1, 2, . . .
(20)
and furthermore there exists 𝑑𝑘 ≥ 0 such that 𝐷𝑛 =
∞, 𝐷𝑛 /𝐷𝑛−1 → 1, and
𝑛
Var ( ∑ 𝑑𝑘 𝜉𝑘 ) ≤
𝑘=1
∑𝑛𝑘=1
𝑑𝑘 ↑
1
󵄨 󵄨
𝑃 (󵄨󵄨󵄨𝑆𝑙𝛼 󵄨󵄨󵄨 ≥ 𝜀𝑙 ) = 𝑂 (log 𝑛) ,
𝑘𝑙𝑝
𝑙
1≤𝑘<𝑙≤𝑛
(29)
(21)
1
1
= 𝑂 (log 𝑛) ,
𝑘𝑙𝑝𝑙 (𝑙 − 𝑘 − 𝑙𝛼 )1/5
1≤𝑘<𝑙≤𝑛
(30)
1 𝑛
∑ 𝑑𝑘 𝜉𝑘 = 1
𝑛→∞𝐷
𝑛 𝑘=1
𝑘<𝑙−𝑙𝛼
1
𝑘𝑙𝑝
𝑙
1≤𝑘<𝑙≤𝑛
∑
a.s.
Remark 7. Let 𝑑𝑘 = 1/𝑘 in (21), and then 𝐷𝑛 =
log 𝑛. Thus, if
𝑘<𝑙−𝑙𝛼
∑
−𝛽
𝑐𝐷𝑛2 (log 𝐷𝑛 ) ,
with some 𝛽 > 1 and positive constant 𝑐, and then
lim
∑
𝑘<𝑙−𝑙𝛼
(22)
∑𝑛𝑘=1
1/𝑘 ∼
1
2
−𝛽
Var ( ∑ 𝜉𝑘 ) ≤ 𝑐(log 𝑛) (log log 𝑛) ,
𝑘
𝑘=1
(23)
(31)
= 𝑂 (log 𝑛) ,
1
𝑘𝑙𝑝
𝑙
1≤𝑘<𝑙≤𝑛
∑
𝑘<𝑙−𝑙𝛼
𝑛
󵄨
󵄨󵄨
𝑎 −𝑏 −𝜀
𝑎 󵄨󵄨
󵄨󵄨
󵄨󵄨Φ ( 𝑙 𝑘 𝑙 ) − Φ ( 𝑙 )󵄨󵄨󵄨
󵄨󵄨
𝑙1/2 󵄨󵄨󵄨
(𝑙 − 𝑘 − 𝑙𝛼 )1/2
󵄨
󵄨󵄨
󵄨󵄨
𝑏 − 𝑎𝑘 + 𝜀𝑙
𝑏𝑙
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨(Φ ( 𝑙
)
−
Φ
(
))
1/2
1/2
󵄨󵄨
󵄨󵄨
𝛼
𝑙
(𝑙 − 𝑘 − 𝑙 )
󵄨
󵄨
= 𝑂 (log 𝑛) .
(32)
4
Journal of Applied Mathematics
The main point in our proof is to verify the condition (23).
We use global central limit theorem with remainders and the
following elementary inequalities:
󵄨
󵄨
󵄨
󵄨󵄨
󵄨󵄨Φ (𝑥) − Φ (𝑦)󵄨󵄨󵄨 ≤ 𝑐 󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 ,
for every 𝑥, 𝑦 ∈ R
Applying Lemma 9, (33), (35), and (36) and noting that
𝜀𝑘 = 𝑘1/2 (log 𝑘)1/3 , we obtain
𝑃 (𝑎𝑙 − 𝑏𝑘 − 𝜀𝑘 ≤ 𝑆𝑙 < 𝑎𝑙 ) + 𝑃 (𝑏𝑙 ≤ 𝑆𝑙 < 𝑏𝑙 − 𝑎𝑘 + 𝜀𝑘 )
(33)
with some constant 𝑐. Moreover, for each 𝑘 > 0, there exists
𝑐1 = 𝑐1 (𝑘), such that
󵄨
󵄨󵄨
󵄨󵄨Φ (𝑥) − Φ (𝑦)󵄨󵄨󵄨
󵄨
󵄨
≥ 𝑐1 󵄨󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 ,
󵄨 󵄨
for every 𝑥, 𝑦 ∈ R, |𝑥| + 󵄨󵄨󵄨𝑦󵄨󵄨󵄨 ≤ 𝑘.
𝑐1 𝑛 ≤ Var (𝑆𝑛 ) =
≤ 𝑐2 𝑛
+ (Φ (
≤
(34)
(35)
for some constant 𝑐1 , 𝑐2 > 0 and sufficiently large 𝑛.
𝑏𝑘 + 𝜀𝑘 −𝑎𝑘 + 𝜀𝑘
𝑏 − 𝑎𝑘 2𝜀𝑘
1
1
+
+ 𝑐 1/5 = 𝑘
+
+ 𝑐 1/5
𝜎𝑙
𝜎𝑙
𝜎𝑙
𝜎𝑙
𝑙
𝑙
≤ 𝑐(
𝑘1/2 (log 𝑘)
𝜀𝑘
𝑘1/2
1
+
+
)
≤
𝑐
(
𝑙1/2 𝑙1/2 𝑙1/5
𝑙1/2
𝑛
𝑙
1 𝑘
𝑘𝑙𝑝𝑙
𝑙=1 𝑘=1
∑∑
,
𝑘 = 1, 2, . . .
=
1
(𝑃 (𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 , 𝑎𝑙 ≤ 𝑆𝑙 < 𝑏𝑙 ) − 𝑝𝑙 𝑝𝑘 )
𝑝𝑙 𝑝𝑘
≤
1
(𝑃 (𝑎𝑙 − 𝑏𝑘 ≤ 𝑆𝑙 − 𝑆𝑘 < 𝑏𝑙 − 𝑎𝑘 ,
𝑝𝑙 𝑝𝑘
1/3
𝑙
𝑛
1/3
𝑛
1/3
(log 𝑙)
𝑙𝑝𝑙
𝑙=1
≤ 𝑐∑
1/3
2
−𝛽
= 𝑂 ((log 𝑛) (log log 𝑛) ) ,
(39)
𝑛 𝑙−1
1 1
1/5
𝑘𝑙𝑝
𝑙 𝑙
𝑙=1 𝑘=1
𝑛
=∑
𝑙=1
1
(𝑃 (𝑎𝑙 − 𝑏𝑘 ≤ 𝑆𝑙 − 𝑆𝑘 < 𝑏𝑙 − 𝑎𝑘 ) − 𝑝𝑙 )
𝑝𝑙
1
(𝑃 (𝑎𝑙 − 𝑏𝑘 − 𝜀𝑘 < 𝑆𝑙 < 𝑏𝑙 − 𝑎𝑘 + 𝜀𝑘 )
𝑝𝑙
󵄨 󵄨
−𝑝𝑙 + 𝑃 (󵄨󵄨󵄨𝑆𝑘 󵄨󵄨󵄨 ≥ 𝜀𝑘 ))
1
(𝑃 (𝑎𝑙 − 𝑏𝑘 − 𝜀𝑘 ≤ 𝑆𝑙 < 𝑎l )
𝑝𝑙
󵄨 󵄨
+ 𝑃 (𝑏𝑙 ≤ 𝑆𝑙 < 𝑏𝑙 − 𝑎𝑘 + 𝜀𝑘 ) +𝑃 (󵄨󵄨󵄨𝑆𝑘 󵄨󵄨󵄨 ≥ 𝜀𝑘 )) .
𝑛
1
1
log 𝑙
≤
𝑐
∑
6/5
6/5
𝑙 𝑝𝑙 𝑘=1 𝑘
𝑙=1 𝑙 𝑝𝑙
1
𝑙−1
∑
1/3
𝑛
(log 𝑙)
≤ 𝑐∑
𝑙𝑝𝑙
𝑙=1
2
−𝛽
= 𝑂 ((log 𝑛) (log log 𝑛) ) .
By Chebyshev’s inequality and the condition of (15) and
(35), we obtain
1
(𝑃 (𝑎𝑙 − 𝑏𝑘 ≤ 𝑆𝑙 − 𝑆𝑘 < 𝑏𝑙 − 𝑎𝑘 )
𝑝𝑙 𝑝𝑘
×𝑃 (𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 ) − 𝑝𝑙 𝑝𝑘 )
≤
1
𝑛
𝑙
(log 𝑙)
(log 𝑙) 1/2
1
≤
𝑐
𝑙
∑
∑
3/2
1/2
3/2
𝑙=1 𝑙 𝑝𝑙 𝑘=1 𝑘
𝑙=1 𝑙 𝑝𝑙
𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 ) − 𝑝𝑙 𝑝𝑘 )
≤
1/3
(log 𝑘)
𝑙1/2
∑∑
Cov (𝛼𝑘 , 𝛼𝑙 )
=
1/2
≤ 𝑐∑
(36)
with some constant 𝑐. Let 1 ≤ 𝑘 < 𝑙 and 𝜀𝑘 = 𝑘1/2 (log 𝑘)1/3 .
If either 𝑝𝑘 = 0 or 𝑝𝑙 = 0, then obviously Cov(𝛼𝑘 , 𝛼𝑙 ) = 0,
and so we may assume that 𝑝𝑘 𝑝𝑙 ≠ 0; then, we have
≤
1
).
𝑙1/5
(38)
(log 𝑘)
≤ ∑ 3/2 ∑
𝑙
𝑝
𝑘1/2
𝑙 𝑘=1
𝑙=1
Proof of Theorem 2. First assume that
𝑏𝑘 − 𝑎𝑘 ≤ 𝑐𝑘
+
By the condition of (15), we have
𝑛
1/2
𝑏𝑙 − 𝑎𝑘 + 𝜀𝑘
𝑏
1
) − Φ ( 𝑙 )) + 𝑐 1/5
𝜎𝑙
𝜎𝑙
𝑙
1/3
Let {𝑋𝑛 , 𝑛 ≥ 1} be a strictly stationary sequence of NA
random variables with 𝜎2 > 0; we can immediately get 𝜎𝑛2 ∼
𝑛𝜎2 , that is,
𝜎𝑛2
𝑎𝑙
𝑎 −𝑏 −𝜀
) − Φ ( 𝑙 𝑘 𝑘 ))
𝜎𝑙
𝜎𝑙
≤ (Φ (
𝑛
(37)
𝑙
1
󵄨 󵄨
𝑃 (󵄨󵄨󵄨𝑆𝑘 󵄨󵄨󵄨 ≥ 𝜀𝑘 )
𝑘𝑙𝑝𝑙
𝑙=1 𝑘=1
∑∑
𝑛
≤∑
𝑙=1
𝑛
𝑛
1 𝑙 Var (𝑆𝑘 )
1 𝑙
1
≤ 𝑐∑
∑
∑
2
2/3
𝑙𝑝𝑙 𝑘=1 𝑘𝜀𝑘
𝑙𝑝
𝑙 𝑘=1 𝑘(log 𝑘)
𝑙=1
1/3
(log 𝑙)
𝑙𝑝𝑙
𝑙=1
≤ 𝑐∑
2
(40)
−𝛽
= 𝑂 ((log 𝑛) (log log 𝑛) ) .
Hence (37)–(40) imply that
𝑛 𝑙−1
Cov (𝛼𝑘 , 𝛼𝑙 )
2
−𝛽
= 𝑂 ((log 𝑛) (log log 𝑛) ) .
𝑘𝑙
𝑙=1 𝑘=1
∑∑
(41)
Journal of Applied Mathematics
5
But Var(𝛼𝑘 ) = 0 if 𝑝𝑘 = 0 and
Var (𝛼𝑘 ) =
we obtain
1 − 𝑝𝑘
1
≤
𝑝𝑘
𝑝𝑘
if 𝑝𝑘 ≠ 0.
lim 𝑃 (−𝑥𝜎𝑘1/2 ≤ 𝑆𝑘 < 0)
(42)
𝑘→∞
= lim 𝑃 (−𝑥𝜎𝑘 ≤ 𝑆𝑘 < 0) = Φ (0) − Φ (−𝑥) ,
Thus
𝑘→∞
𝑛
Var (𝛼𝑘 )
𝑘2
𝑘=1
𝑘→∞
= lim 𝑃 (0 ≤ 𝑆𝑘 < 𝑥𝜎𝑘 ) = Φ (𝑥) − Φ (0) .
1/3
(log 𝑘)
1
≤ ∑ 2 ≤ ∑
𝑘 𝑝𝑘 1≤𝑘≤𝑛 𝑘𝑝𝑘
1≤𝑘≤𝑛
𝑝𝑘 ≠ 0
(43)
𝑝𝑘 ≠ 0
2
𝑘→∞
Applying the almost sure central limit theorem for NA
random variables (5), that is,
−𝛽
1 𝑛 1
∑ 𝐼 {𝑆𝑘 ≤ 𝑥𝜎𝑘1/2 } = Φ (𝑥)
𝑛 → ∞ log 𝑛
𝑘
𝑘=1
= 𝑂 ((log 𝑛) (log log 𝑛) ) .
lim
Equations (41) and (43) together imply that
𝑛
𝛼
2
−𝛽
Var ( ∑ 𝑘 ) = 𝑂 ((log 𝑛) (log log 𝑛) ) ,
𝑘
𝑘=1
as 𝑛 → ∞.
Lemma 8, and (48), we have
𝐼 {𝑆𝑘 < −𝑥𝜎𝑘1/2 }
1 𝑛
∑
1/2 ≤ 𝑆 < 0)
𝑛 → ∞ log 𝑛
𝑘
𝑘=1 𝑘𝑃 (−𝑥𝜎𝑘
lim
Hence applying Remark 7, our theorem is proved under
the restricting condition (36).
Now we drop the restricting condition (36). Fix 𝑥 > 0 and
define
=
̃𝑏 = min (𝑏 , 𝑥𝜎𝑘1/2 ) ,
𝑘
𝑘
(45)
=
𝑝̃𝑘 = 𝑃 (̃
𝑎𝑘 ≤ 𝑆𝑘 < ̃𝑏𝑘 ) ,
where 𝜎 is defined by (3).
Clearly 𝑝̃𝑘 ≤ 𝑝𝑘 , and so assuming 𝑝̃𝑘 ≠ 0; then, also we
have 𝑝𝑘 ≠ 0, and thus
1
𝛼𝑘 = 𝐼 {𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 }
𝑝𝑘
a.s.
1 𝑛 𝛼̃𝑘
∑
= 1 a.s.,
𝑛 → ∞ log 𝑛
𝑘
𝑘=1
(51)
𝐼 {̃
𝑎𝑘 ≤ 𝑆𝑘 < ̃𝑏𝑘 }
{
{
, if 𝑝̃𝑘 ≠ 0,
𝛼̃𝑘 = {
𝑝̃𝑘
{
if 𝑝̃𝑘 = 0.
{1,
(52)
Equations (46) and (50)–(51) together imply that
(46)
lim sup
1
(𝐼 {𝑎𝑘 ≤ 𝑆𝑘 < 𝑎̃𝑘 } + 𝐼 {̃𝑏𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 })
+
𝑝𝑘
𝑛→∞
+
𝐼 {𝑆𝑘 ≥ 𝑥𝜎𝑘1/2 }
𝑃 (0 ≤ 𝑆𝑘 < 𝑥𝜎𝑘1/2 )
1 𝑛 𝛼𝑘
1 − Φ (𝑥)
∑
≤1+2
log 𝑛 𝑘=1 𝑘
Φ (𝑥) − Φ (0)
a.s.
(53)
On the other hand, if 𝑝̃𝑘 ≠ 0, then we have
1
𝐼 {̃
𝑎𝑘 ≤ 𝑆𝑘 < ̃𝑏𝑘 }
𝑝̃𝑘
𝑃 (−𝑥𝜎𝑘1/2 ≤ 𝑆𝑘 < 0)
1 − Φ (𝑥)
Φ (𝑥) − Φ (0)
where
1
𝐼 {̃
𝑎𝑘 ≤ 𝑆𝑘 < ̃𝑏𝑘 }
̃
𝑝𝑘
𝐼 {𝑆𝑘 < −𝑥𝜎𝑘1/2 }
(50)
lim
1
(𝐼 {̃
𝑎𝑘 ≤ 𝑆𝑘 < ̃𝑏𝑘 }
𝑝𝑘
+
a.s.,
Since 𝑎̃𝑘 and ̃𝑏𝑘 satisfy (36), we get
+𝐼 {𝑎𝑘 ≤ 𝑆𝑘 < 𝑎̃𝑘 } + 𝐼 {̃𝑏𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 })
≤
Φ (−𝑥)
Φ (0) − Φ (−𝑥)
𝐼 {𝑆𝑘 > 𝑥𝜎𝑘1/2 }
1 𝑛
lim
∑
1/2
𝑛 → ∞ log 𝑛
𝑘=1 𝑘𝑃 (0 ≤ 𝑆𝑘 < 𝑥𝜎𝑘 )
𝑎̃𝑘 = max (𝑎𝑘 , −𝑥𝜎𝑘1/2 ) ,
≤
a.s. ∀𝑥 ∈ R,
(49)
(44)
=
(48)
lim 𝑃 (0 ≤ 𝑆𝑘 < 𝑥𝜎𝑘1/2 )
∑
1
𝐼 {𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 }
𝑝𝑘
.
By (35) and the central limit theorem for NA random
variables (4), that is,
󵄨󵄨
󵄨󵄨
𝑆
󵄨
󵄨
sup 󵄨󵄨󵄨𝑃 ( 𝑛 < 𝑥) − Φ (𝑥)󵄨󵄨󵄨 = 𝑜 (1) ,
(47)
𝜎𝑛
𝑥∈𝑅 󵄨󵄨
󵄨󵄨
≥
𝑝 − 𝑝̃𝑘
1
𝐼 {̃
𝑎𝑘 ≤ 𝑆𝑘 < ̃𝑏𝑘 } (1 − 𝑘
)
𝑝̃𝑘
𝑝𝑘
≥ 𝛼̃𝑘 (1 −
𝑃 (𝑆𝑘 < −𝜎𝑥𝑘1/2 ) + 𝑃 (𝑆𝑘 > 𝜎𝑥𝑘1/2 )
),
min{𝑃 (0 ≤ 𝑆𝑘 < 𝜎𝑥𝑘1/2 ) , 𝑃 (−𝜎𝑥𝑘1/2 ≤ 𝑆𝑘 < 0)}
(54)
6
Journal of Applied Mathematics
and by the central limit theorem,
lim
𝑃 (𝑆𝑘 < −𝜎𝑥𝑘
𝑘→∞
min {𝑃 (0 ≤ 𝑆𝑘 <
Applying Lemma 9, (33), and (35), we obtain
1/2
1/2
) + 𝑃 (𝑆𝑘 > 𝜎𝑥𝑘
1/2
𝜎𝑥𝑘 ) , 𝑃 (−𝜎𝑥𝑘1/2 ≤
)
𝑃 (𝑎𝑙 − 𝑏𝑘 − 𝜀𝑙 < 𝑆𝑙−𝑘−𝑙𝛼 < 𝑏𝑙 − 𝑎𝑘 + 𝜀𝑙 ) − 𝑝𝑙
𝑆𝑘 < 0)}
1 − Φ (𝑥)
.
=1−2
Φ (𝑥) − Φ (0)
≤ (Φ (
(55)
𝑏𝑙 − 𝑎𝑘 + 𝜀𝑙
(𝑙 − 𝑘 −
+ (Φ (
Applying Lemma 8, (51) and (54) imply that
lim inf
𝑛→∞
1 𝑛 𝛼𝑘
1 − Φ (𝑥)
≥1−2
∑
log 𝑛 𝑘=1 𝑘
Φ (𝑥) − Φ (0)
a.s.,
1 𝑛 𝛼𝑘
1 − Φ (𝑥)
≥ lim sup
∑
Φ (𝑥) − Φ (0)
𝑛 → ∞ log 𝑛 𝑘=1 𝑘
≥ lim inf
𝑛→∞
≥1−2
1 ≥ lim sup
𝑛→∞
1 − Φ (𝑥)
Φ (𝑥) − Φ (0)
(57)
1
(𝑙 − 𝑘 − 𝑙𝛼 )1/5
))
(61)
).
Cov (𝛼𝑘 , 𝛼𝑙 )
= 𝑂 (log 𝑛) ,
𝑘𝑙
1≤𝑘<𝑙≤𝑛
∑
𝛼
1
∑ 𝑘
log 𝑛 𝑘=1 𝑘
because 𝑙 − 𝑙𝛼 < 𝑘 < 𝑙, 𝑙𝛾 − (𝑙 − 𝑙𝛼 )𝛾 → 0 as 𝑙 → ∞ for
𝛾 < 1, 𝛼 < 1.
But Var(𝛼𝑘 ) = 0 if 𝑝𝑘 = 0 and
(58)
Var (𝛼𝑘 ) =
a.s.
1 − 𝑝𝑘
1
≤
𝑝𝑘
𝑝𝑘
if 𝑝𝑘 ≠ 0.
(64)
Then
𝑛
(59)
This completes the proof of Theorem 2.
Proof of Theorem 5. Let 𝑘 < 𝑙 − 𝑙𝛼 , 1 ≤ 𝑘 < 𝑙, and 𝜀𝑙 = 𝑙3𝛼/2 ;
we have
Cov (𝛼𝑘 , 𝛼𝑙 )
1
(𝑃 (𝑎𝑙 − 𝑏𝑘 ≤ 𝑆𝑙 − 𝑆𝑘+𝑙𝛼 + 𝑆𝑘+𝑙𝛼 − 𝑆𝑘
𝑝𝑙 𝑝𝑘
< 𝑏𝑙 − 𝑎𝑘 , 𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 ) − 𝑝𝑙 𝑝𝑘 )
1
(𝑃 (𝑎𝑙 − 𝑏𝑘 ≤ 𝑆𝑙 − 𝑆𝑘+𝑙𝛼 + 𝑆𝑘+𝑙𝛼 − 𝑆𝑘
𝑝𝑙 𝑝𝑘
< 𝑏𝑙 − 𝑎𝑘 ) 𝑃 (𝑎𝑘 ≤ 𝑆𝑘 < 𝑏𝑘 ) − 𝑝𝑙 𝑝𝑘 ) (60)
1
(𝑃 (𝑎𝑙 − 𝑏𝑘 − 𝜀𝑙 < 𝑆𝑙 − 𝑆𝑘+𝑙𝛼 < 𝑏𝑙 − 𝑎𝑘 + 𝜀𝑙 )
𝑝𝑙
󵄨
󵄨
−𝑝𝑙 + 𝑃 (󵄨󵄨󵄨𝑆𝑘+𝑙𝛼 − 𝑆𝑘 󵄨󵄨󵄨 ≥ 𝜀𝑙 ))
(63)
𝑘>𝑙−𝑙𝛼
𝑛
1
(𝑃 (𝑎𝑙 − 𝑏𝑘 − 𝜀𝑙 < 𝑆𝑙−𝑘−𝑙𝛼 < 𝑏𝑙 − 𝑎𝑘 + 𝜀𝑙 )
𝑝𝑙
󵄨 󵄨
−𝑝𝑙 + 𝑃 (󵄨󵄨󵄨𝑆𝑙𝛼 󵄨󵄨󵄨 ≥ 𝜀𝑙 )) .
(62)
On the other hand,
a.s.
1 𝑛 𝛼𝑘
lim
∑
= 1 a.s.
𝑛 → ∞ log 𝑛
𝑘
𝑘=1
≤
(𝑙 − 𝑘 −
𝑙𝛼 )1/2
𝑘<𝑙−𝑙𝛼
Thus
≤
+
𝑎𝑙 − 𝑏𝑘 − 𝜀𝑙
− Φ(
Cov (𝛼𝑘 , 𝛼𝑙 )
= 𝑂 (log 𝑛) .
𝑘𝑙
1≤𝑘<𝑙≤𝑛
1 𝑛 𝛼𝑘
∑
log 𝑛 𝑘=1 𝑘
1 𝑛 𝛼𝑘
≥ lim inf
≥1
∑
𝑛 → ∞ log 𝑛
𝑘
𝑘=1
≤
1
𝑙1/5
𝑏𝑙
))
1/2
𝑙
∑
By the arbitrariness of 𝑥, let 𝑥 → ∞ in (57); we have
≤
𝑎𝑙
)
𝑙1/2
) − Φ(
Hence Lemma 11, (60), and (61) imply that
and hence
1+2
+ 𝑐(
(56)
𝑙𝛼 )1/2
𝑛
Var (𝛼𝑘 )
1
1
≤
≤
= 𝑂 (log 𝑛) .
∑
∑
2
2𝑝
5/2−𝛼
𝑘
𝑘
𝑘
𝑘
𝑘=1
1≤𝑘≤𝑛
𝑘=1
∑
(65)
𝑝𝑘 ≠ 0
Noting that
𝑛
𝛼𝑘
)
𝑘
𝑘=1
Var ( ∑
𝑛
Var (𝛼𝑘 )
𝑘2
𝑘=1
=∑
(66)
Cov (𝛼𝑘 , 𝛼𝑙 )
Cov (𝛼𝑘 , 𝛼𝑙 )
+2 ∑
,
𝑘𝑙
𝑘𝑙
1≤𝑘<𝑙≤𝑛
1≤𝑘<𝑙≤𝑛
+2 ∑
𝑘>𝑙−𝑙𝛼
𝑘<𝑙−𝑙𝛼
thus (62)–(66) imply that
𝑛
𝛼𝑘
) = 𝑂 (log 𝑛) ,
𝑘
𝑘=1
Var ( ∑
as 𝑛 󳨀→ ∞.
(67)
Hence applying Remark 7, we have
1 𝑛 𝛼𝑘
∑
= 1 a.s.
𝑛 → ∞ log 𝑛
𝑘
𝑘=1
lim
This completes the proof of Theorem 5.
(68)
Journal of Applied Mathematics
7
Appendix
∑𝑛𝑘=1
Proof of Lemma 6. Let 𝜇𝑛 =
Proof of Lemma 10. Applying Lemma 9, (33), and noting the
conditions of (19) and 0 < 𝛼 < 1/7, we get
𝑑𝑘 𝜉𝑘 , and then ∀𝜀 > 0
𝑝𝑘 = 𝑃 (𝑎𝑘 ≤ 𝑆𝑘 ≤ 𝑏𝑘 )
󵄨󵄨 𝜇
Var (𝜇𝑛 /𝐷𝑛 )
𝐸𝜇 󵄨󵄨󵄨
−𝛽
󵄨
≤ 𝑐(log 𝐷𝑛 ) . (A.1)
𝑃 (󵄨󵄨󵄨 𝑛 − 𝑛 󵄨󵄨󵄨 ≥ 𝜀) ≤
󵄨󵄨 𝐷𝑛 𝐷𝑛 󵄨󵄨
𝜀2
≤ (Φ (
𝑟
Let 𝑟 < 1, 𝑟𝛽 > 1, and 𝑛𝑘 = inf{𝑛𝑘 , 𝐷𝑛𝑘 ≥ exp(𝑘 )}, and
then 𝐷𝑛𝑘 ≥ exp(𝑘𝑟 ), 𝐷𝑛𝑘 −1 < exp(𝑘𝑟 ), for 𝐷𝑛 ∼ 𝐷𝑛−1 ; we get
1≤
𝐷𝑛𝑘
∼
exp (𝑘𝑟 )
𝐷𝑛𝑘 −1
< 1 󳨀→ 1,
exp (𝑘𝑟 )
(A.2)
𝐷𝑛𝑘 ∼ exp (𝑘𝑟 ) ,
𝐷𝑛𝑘−1
∼
≥ (Φ (
exp (𝑘 )
exp ((𝑘 − 1)𝑟 )
≥ 𝑐(
𝑟
1
1
= exp (𝑘𝑟 [1 − (1 − ) ]) ∼ exp (𝑘𝑟 ⋅ 𝑟 ⋅ )
𝑘
𝑘
= exp (𝑟 ⋅ 𝑘
On account of 𝑟 < 1, then
𝐷𝑛𝑘−1
∼ exp (𝑟 ⋅ 𝑘𝑟−1 ) 󳨀→ 1,
𝑘=1
Proof of Lemma 11. By Lemma 10, Chebyshev’s inequality,
and noting the condition of 0 < 𝛼 < 1/7, we have
1
󵄨 󵄨
𝑃 (󵄨󵄨󵄨𝑆𝑙𝛼 󵄨󵄨󵄨 ≥ 𝜀𝑙 )
𝑘𝑙𝑝
𝑙
1≤𝑘<𝑙≤𝑛
𝑘<𝑙−𝑙𝛼
𝛼
(A.5)
≤
(log 𝐷𝑛 )
−𝛽
𝐷𝑛𝑘
𝐸𝜇𝑛𝑘
log (𝑙 − 𝑙𝛼 )
= 𝑂 (log𝑛) .
𝑙1+𝛼
𝑙=1
𝑛
≤ 𝑐∑
󳨀→ 0 a.s.
𝐷𝑛𝑘
(A.6)
𝐷𝑛𝑘
1
1
𝑘𝑙𝑝𝑙 (𝑙 − 𝑘 − 𝑙𝛼 )1/5
1≤𝑘<𝑙≤𝑛
𝑘<𝑙−𝑙𝛼
𝑛
∑ 𝑘 𝑑𝑘
= 𝑘=1
= 1,
𝐷𝑛𝑘
(A.7)
𝑛
≤ 𝑐∑
𝑙=1
thus
𝜇𝑛𝑘
𝐷𝑛𝑘
󳨀→ 1,
a.s.
𝐷𝑛𝑘 −1 𝜇𝑛𝑘−1
𝐷𝑛𝑘 𝐷𝑛𝑘−1
≤
𝜇𝑛 𝐷𝑛𝑘
𝜇𝑛
≤ 𝑘
󳨀→ 1
𝐷𝑛 𝐷𝑛𝑘 𝐷𝑛𝑘−1
1
1
( ∑
𝑙1−𝛼 1≤𝑘<(𝑙−𝑙𝛼 )/2 𝑘(𝑙 − 𝑙𝛼 − 𝑘)1/5
(A.8)
Now for 𝑛𝑘−1 ≤ 𝑛 < 𝑛𝑘 , for 𝐷𝑛 ↑ ∞, 𝐷𝑛𝑘 /𝐷𝑛𝑘−1 → 1, and by
𝜉𝑘 ≥ 0, then 𝜇𝑛 ↑, and we have
1 ←󳨀
It proves (29). By Lemma 10 and 0 < 𝛼 < 1/7, we get
∑
Since
𝐸𝜇𝑛𝑘
(A.13)
𝑘<𝑙−𝑙
1
< ∞.
𝑟𝛽
𝑘=1 𝑘
≤ 𝑐∑
By the Borel-Cantelli lemma,
−
𝑛
1 Var (𝑆𝑙𝛼 )
1 𝑙−𝑙 1
≤ 𝑐∑ 1+𝛼 ∑
3𝛼
𝑘𝑙𝑝𝑙 𝑙
𝑙
𝑘
1≤𝑘<𝑙≤𝑛
𝑙=1
𝑘=1
∑
𝛼
∞
1
𝜇𝑛𝑘
𝑘
1
+ 1/5 ) ≥ 𝑐1󸀠 𝑘−𝛼 .
𝑘1/2
𝑘
∑
as 𝑘 󳨀→ ∞,
󵄨󵄨 𝜇
𝐸𝜇𝑛𝑘 󵄨󵄨󵄨󵄨
󵄨 𝑛
󵄨󵄨 ≥ 𝜀)
∑ 𝑃 (󵄨󵄨󵄨󵄨 𝑘 −
𝐷𝑛𝑘 󵄨󵄨󵄨
󵄨󵄨 𝐷𝑛𝑘
𝑘=1
∞
(A.12)
1/2−𝛼
Thus Lemma 10 immediately follows from (A.11) and (A.12).
∞
≤ 𝑐∑
𝑏𝑘
𝑎𝑘
1
) − Φ ( 1/2
)) − 𝑐 1/5
1/2
𝑘
𝑘
𝑘
).
(A.4)
𝐷𝑛𝑘
𝑘1/2−𝛼
1
+ 1/5 ) ≤ 𝑐2󸀠 𝑘−𝛼 .
1/2
𝑘
𝑘
𝑝𝑘 = 𝑃 (𝑎𝑘 ≤ 𝑆𝑘 ≤ 𝑏𝑘 )
(A.3)
𝑟
𝑟−1
(A.11)
Applying Lemma 9, (34), and noting the conditions of (19)
and 0 < 𝛼 < 1/7, we have
that is,
and thus
𝐷𝑛𝑘
≤ 𝑐(
𝑏𝑘
𝑎𝑘
1
) − Φ ( 1/2
)) + 𝑐 1/5
𝑘1/2
𝑘
𝑘
a.s.
(A.9)
+
𝑛
≤ 𝑐∑
𝑙=1
+
This completes the proof of Lemma 6.
(A.10)
𝑛
≤ 𝑐∑
𝑙=1
(𝑙−𝑙𝛼 )/2<𝑘<𝑙−𝑙𝛼 𝑘(𝑙
1
−
𝑙𝛼
− 𝑘)1/5
1
1
1
(
∑
𝑙1−𝛼 (𝑙 − 𝑙𝛼 )1/5 1≤𝑘<(𝑙−𝑙𝛼 )/2 𝑘
hence
𝜇𝑛
󳨀→ 1 a.s.
𝐷𝑛
∑
log 𝑙
𝑙1−𝛼+1/5
1
1
)
∑
𝑙 − 𝑙𝛼 1≤𝑘<(𝑙−𝑙𝛼 )/2 𝑘1/5
𝑛
1
≤ 𝑐∑ = 𝑂 (log 𝑛) .
𝑙
𝑙=1
)
(A.14)
8
Journal of Applied Mathematics
It proves (30). Applying Lemma 9, (33), (35), 𝜀𝑙 = 𝑙3𝛼/2 , and
noting the condition of 0 < 𝛼 < 1/7, we obtain
Noting that 0 < 𝛼 < 1/7, we deduce
1
Σ3 ≤ 𝑐 ∑
1−(7/2)𝛼
1/2
𝑘 (𝑙 − 𝑙𝛼 − 𝑘)1/2
1≤𝑘<𝑙≤𝑛 𝑙
𝑘<𝑙−𝑙𝛼
󵄨
󵄨
𝑎 −𝑏 −𝜀
𝑎 󵄨󵄨
1 󵄨󵄨󵄨
󵄨󵄨Φ ( 𝑙 𝑘 𝑙 ) − Φ ( 𝑙 )󵄨󵄨󵄨
∑
𝑘𝑙𝑝𝑙 󵄨󵄨󵄨
𝑙1/2 󵄨󵄨󵄨
(𝑙 − 𝑘 − 𝑙𝛼 )1/2
1≤𝑘<𝑙≤𝑛
𝛼
1
1−(7/2)𝛼
𝑙
1≤𝑙≤𝑛
≤𝑐 ∑
𝑘<𝑙−𝑙
−𝑎𝑙
1
1
(
− )
𝛼 )1/2
√
𝑘𝑙𝑝
𝑙
−
𝑘
−
𝑙
(𝑙
𝑙
1≤𝑘<𝑙≤𝑛
×(
≤𝑐 ∑
1
(𝑙 −
𝛼
𝑘<𝑙−𝑙
(A.15)
𝑏𝑘
1
+𝑐 ∑
𝑘𝑙𝑝𝑙 (𝑙 − 𝑘 − 𝑙𝛼 )1/2
1≤𝑘<𝑙≤𝑛
≤𝑐 ∑
1≤𝑙≤𝑛
𝑙𝛼 )1/2
1
1
1
)
+
∑
𝛼
1/2
𝑘 (𝑙 − 𝑙 ) 1≤𝑘<(𝑙−𝑙𝛼 )/2 𝑘
1≤𝑘<(𝑙−𝑙𝛼 )/2
log 𝑙
𝑙3/2−(7/2)𝛼
∑
(A.17)
𝛼
𝑘<𝑙−𝑙
It proves (31). The proof of (32) is similar to the proof of (31).
This completes the proof of Lemma 11.
𝜀𝑙
1
+𝑐 ∑
:= Σ1 + Σ2 + Σ3 .
𝑘𝑙𝑝𝑙 (𝑙 − 𝑘 − 𝑙𝛼 )1/2
1≤𝑘<𝑙≤𝑛
𝑘<𝑙−𝑙𝛼
Acknowledgments
The authors are very grateful to the academic editor, professor Ying Hu, and the two anonymous reviewers for
their valuable comments and helpful suggestions, which
significantly contributed to improving the quality of this
paper. This work is jointly supported by National Natural
Science Foundation of China (11061012,71271210), Project
Supported by Program to Sponsor Teams for Innovation in
the Construction of Talent Highlands in Guangxi Institutions
of Higher Learning ((2011)47), the Guangxi China Science
Foundation (2013GXNSFDA019001).
Now applying the same procedure as before, we have
−𝑎𝑙
1
𝑘
(
+
)
1/2
1−𝛼
𝛼
𝑘𝑙𝑝
𝑙 (𝑙 − 𝑘 − 𝑙 )
𝑙(𝑙 − 𝑘 − 𝑙𝛼 )1/2
𝑙
1≤𝑘<𝑙≤𝑛
Σ1 ≤ 𝑐 ∑
𝑘<𝑙−𝑙𝛼
≤𝑐 ∑
1≤𝑙≤𝑛
1
𝑙3/2−𝛼
References
1
1
1
×(
)
+
∑
∑
𝛼
1/2
1/2
𝛼
𝑘
𝑙
−
𝑙
𝑘
(l − 𝑙 ) 1≤𝑘<(𝑙−𝑙𝛼 )/2
1≤𝑘<(𝑙−𝑙𝛼 )/2
1
1
1
∑
3/2
𝑙
𝛼
−
𝑘
−
𝑙𝛼 )1/2
(𝑙
1≤𝑙≤𝑛
1≤𝑘<𝑙−𝑙
+𝑐 ∑
log 𝑙
1
≤𝑐 ∑
= 𝑂 (log 𝑛) ,
2−𝛼
𝑙
𝑙
1≤𝑙≤𝑛
1≤𝑙≤𝑛
≤𝑐 ∑
Σ2 ≤ 𝑐 ∑
1
1≤𝑘<𝑙≤𝑛
𝑘<𝑙−𝑙𝛼
≤𝑐 ∑
1≤𝑙≤𝑛
𝑙1−𝛼 𝑘1/2+𝛼 (𝑙
1
𝑙1−𝛼
(
1≤𝑙≤𝑛
1
𝑙1−𝛼
(
− 𝑙𝛼 − 𝑘)1/2
1
(𝑙 −
𝑙𝛼 )1/2
+
≤𝑐 ∑
1
= 𝑂 (log 𝑛) .
𝑙
1≤𝑙≤𝑛
≤𝑐 ∑
𝑙𝛼 )1/2+𝛼
1
(𝑙 −
1≤𝑘<(𝑙−𝑙𝛼 )/2
1
(𝑙 −
𝑙𝛼 )1/2
1
∑
𝑘1/2+𝛼
1
)
1/2
𝑘
1≤𝑘<(𝑙−𝑙𝛼 )/2
𝑙1/2−𝛼 +
∑
1
(𝑙 −
𝑙𝛼 )1/2+𝛼
1/2
(𝑙 − 𝑙𝛼 )
)
1
= 𝑂 (log 𝑛) .
𝑙
1≤𝑙≤𝑛
≤𝑐 ∑
(A.16)
[1] K. Joag-Dev and F. Proschan, “Negative association of random
variables, with applications,” The Annals of Statistics, vol. 11, no.
1, pp. 286–295, 1983.
[2] Q. M. Shao, “A comparison theorem on moment inequalities
between negatively associated and independent random variables,” Journal of Theoretical Probability, vol. 13, no. 2, pp. 343–
356, 2000.
[3] Q. Y. Wu and Y. Y. Jiang, “A law of the iterated logarithm of
partial sums for NA random variables,” Journal of the Korean
Statistical Society, vol. 39, no. 2, pp. 199–206, 2010.
[4] C. M. Newman, “Asymptotic independence and limit theorems
for positively and negatively dependent random variables,”
in Inequalities in Statistics and Probability, vol. 5 of Lecture
Notes-Monograph Series, pp. 127–140, Institute of Mathematical
Statistics, Hayward, Calif, USA, 1984.
[5] P. Matuła, “Some limit theorems for negatively dependent
random variables,” Yokohama Mathematical Journal, vol. 41, no.
2, pp. 163–173, 1994.
[6] P. Matuła, “On almost sure limit theorems for positively dependent random variables,” Statistics & Probability Letters, vol. 74,
no. 1, pp. 59–66, 2005.
[7] Q. Y. Wu, “An almost sure central limit theorem for the weight
function sequences of NA random variables,” Proceedings—
Mathematical Sciences, vol. 121, no. 3, pp. 369–377, 2011.
[8] G. A. Brosamler, “An almost everywhere central limit theorem,”
Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 3, pp. 561–574, 1988.
Journal of Applied Mathematics
[9] P. Schatte, “On strong versions of the central limit theorem,”
Mathematische Nachrichten, vol. 137, pp. 249–256, 1988.
[10] M. Peligrad and Q. M. Shao, “A note on the almost sure
central limit theorem for weakly dependent random variables,”
Statistics & Probability Letters, vol. 22, no. 2, pp. 131–136, 1995.
[11] P. Matuła, “On the almost sure central limit theorem for
associated random variables,” Probability and Mathematical
Statistics, vol. 18, no. 2, pp. 411–416, 1998.
[12] E. Csáki, A. Földes, and P. Révész, “On almost sure local and
global central limit theorems,” Probability Theory and Related
Fields, vol. 97, no. 3, pp. 321–337, 1993.
[13] G. Hurelbaatar, “An almost sure local and global central limit
theorem for weakly dependent random variables,” Annales
Universitatis Scientiarum Budapestinensis de Rolando Eötvös
Nominatae, Sectio Mathematica, vol. 38, pp. 109–126, 1995.
[14] Z. C. Weng, Z. C. Peng, and S. Nadarajah, “The almost sure
local central limit theorem for the product of partial sums,”
Proceedings—Mathematical Sciences, vol. 121, no. 2, pp. 217–228,
2011.
[15] I. Berkes and E. Csáki, “A universal result in almost sure central
limit theory,” Stochastic Processes and their Applications, vol. 94,
no. 1, pp. 105–134, 2001.
[16] A. Földes, “Some of my favorite results with Endre Csáki and Pál
Révész: a survey,” Periodica Mathematica Hungarica, vol. 50, no.
1-2, pp. 117–134, 2005.
[17] J. M. Pan, “On the convergence rate in the central limit theorem
for negatively associated sequences,” Chinese Journal of Applied
Probability and Statistics, vol. 13, no. 2, pp. 183–192, 1997.
9
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