- Email: [email protected]
On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables
Acta Mathematica Scientia 2007,27B(2):283–290 http://actams.wipm.ac.cn
ON THE LIMITING BEHAVIOR OF THE MAXIMUM PARTIAL SUMS FOR ARRAYS OF ROWWISE NA RANDOM VARIABLES∗ Gan Shixin (
)
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China E-mail: [email protected]
Chen Pingyan (
)
Department of Mathematics, Jinan University, Guangzhou 510630, China
Abstract Let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables and {an , n ≥ 1} a sequence of constants with 0 < an ↑ ∞. The limiting behavior of maximum partial sums
k 1 max | Xni | an 1≤k≤n i=1
is investigated and some new results are obtained. The
results extend and improve the corresponding theorems of rowwise independent random variable arrays by Hu and Taylor [1] and Hu and Chang [2]. Key words NA random variable, maximum partial sum, complete convergence, convergence in probability 2000 MR Subject Classification
1
60F15, 60G50
Introduction and Main Results
Let (Ω, F , P ) be a probability space and {Xn , n ≥ 1} a sequence of random variables defined on (Ω, F , P ). We start with definitions. A finite family of random variables {Xi , 1 ≤ i ≤ n} is said to be negatively associated(abbreviated to NA) if for any disjoint subsets A and B of {1, 2, · · · , n} and any real coordinatewise nondecreasing functions f on RA and g on RB , Cov (f (Xi , i ∈ A), g(Xj , j ∈ B)) ≤ 0,
(1.1)
whenever the covariance exists. An infinite family of random variables {Xi , 1 ≤ i < ∞} is NA if every finite subfamily is NA. This concept was introduced by Joag–Dev and Proschan [3]. They also pointed out and proved in their paper that a number of well-known multivariate distributions process the NA property. Now people know NA random variables have wide applications in reliability theory and multivariate statistical analysis. Recently Su et al. [4] showed that NA structure plays an important role in risk management. Because of these reasons ∗ Received
February 10, 2004
284
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Vol.27 Ser.B
the notions of NA random variables have received more and more attention in recent years. A great number of papers for NA random variables are now in literature. We refer to Joag–Dev and Proschan [3] for fundamental properties, Newman [5] for the central limit theorem, Matula [6] for the three series theorem, Shao and Su [7] for the law of iterated logarithm, Shao [8] for moment inequalities, Liu et al. [9] for the H` ajeck–R`enyi inequality and Barbour et al. [10] for Poison approximation. Recently Shao [8] showed the following important moment inequalities: let p > 1, {Xi , 1 ≤ i ≤ n} be a sequence of NA mean zero random variables with E|Xi |p < ∞ for every 1 ≤ i ≤ n. Then n k Xi |p ≤ 23−p E|Xi |p for 1 < p ≤ 2, (1.2) E max | 1≤k≤n
and E max | 1≤k≤n
k
i=1
i=1
p
p
Xi | ≤ 2(15p/ ln p)
n n p EXi2 ) 2 + E|Xi |p (
i=1
i=1
for p > 2.
(1.3)
i=1
For a triangular array of rowwise independent random variables {Xni , 1 ≤ i ≤ n, n ≥ 1} , a sequence of positive real numbers {an , n ≥ 1} with an ↑ ∞, and Ψ(t) a positive, even function such that ψ(|t|) Ψ(|t|) ↑ and ↓ as |t| ↑, (1.4) p |t| |t|p+1 for some nonnegative integer p. Conditions are given as EXni = 0, 1 ≤ i ≤ n, n ≥ 1, ∞ n
E
n=1 i=1
n ∞ n=1
i=1
Ψ(Xni ) < ∞, Ψ(an )
E
Xni an
(1.5) (1.6)
2 2k < ∞,
(1.7)
where k is a positive integer. In 1997 Hu and Taylor [1] proved the following theorems: Theorem A Let {Xni , 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise independent random variables and let Ψ(t) satisfy (1.4) for some integer p > 2. Then conditions (1.5), (1.6), and (1.7) imply n 1 Xni → 0 a.s.. (1.8) an i=1 Theorem B Let {Xni , 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise independent random variables and let Ψ(t) satisfy (1.4) for p = 1. Then conditions (1.5), (1.6) imply (1.8). Theorem C Let {Xni , 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise independent random variables and let Ψ(t) satisfy (1.4) for p = 0. Then condition (1.6) implies (1.8). The main purpose of this article is to utilize Shao’s inequalities to extend Theorem A and Theorem B to the case of NA random variables and Theorem C to the case of any random variables. It is worth to point out that our conclusions are much stronger and conditions are more general and much weaker. To prove our main results we need the following lemma:
No.2
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Lemma 1 Nondecreasing functions defined on disjoint subsets of a set of negatively associated random variables are negatively associated (see [3]). A sequence of random variables {Un , n ≥ 1} is said to converge completely to a constant a if for any ε > 0, ∞ P (|Un − a| > ε) < ∞. (1.9) n=1
In this case we write Un → a completely. This notion was given firstly by Hsu and Robbins [11]. Obviously the complete convergence implies the almost sure convergence. In this article, c always stands for a positive constant, which may differ from one place to another. Now, let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables and {an , n ≥ 1} a sequence of positive real numbers with an ↑ ∞. {Ψn (t), n ≥ 1} is a sequence of nonnegative even functions such that Ψn (t) > 0 as t > 0 and Ψn (|t|) Ψn (|t|) ↑ and ↓ as |t| ↑ . |t| |t|p
(1.10)
Conditions (1.11)–(1.13) are given as EXni = 0, 1 ≤ i ≤ n, n ≥ 1, ∞ n n=1 i=1
E
Ψi (Xni ) < ∞, Ψi (an )
n s ∞ E|Xni |r < ∞, arn n=1 i=1
(1.11) (1.12)
(1.13)
where 0 < r ≤ 2, s > 0. Theorem 1 Let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables, {an , n ≥ 1} a sequence of positive real numbers with an ↑ ∞ and {Ψn (t), n ≥ 1} a sequence of nonnegative even functions such that Ψn (t) > 0 for t > 0 and Ψn (|t|) Ψn (|t|) ↑ and ↓ as |t| ↑, |t| |t|p where 1 < p ≤ 2. Then Conditions (1.11) and (1.12) imply k 1 max | Xni | → 0 completely. an 1≤k≤n i=1
(1.14)
Theorem 2 Let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables, {an , n ≥ 1} a sequence of positive real numbers with an ↑ ∞ and {Ψn (t), n ≥ 1} a sequence of nonnegative even functions such that Ψn (t) > 0 for t > 0 and Ψn (|t|) Ψn (|t|) ↑ and ↓ as |t| ↑, |t| |t|p where p > 2. Then conditions (1.11), (1.12), and (1.13) imply (1.14).
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Theorem 3 Let {Xni , 1 ≤ n, i < ∞} be an array of any random variables, {an , n ≥ 1} a sequence of positive real numbers with an ↑ ∞ and {Ψn (t), n ≥ 1} a sequence of nonnegative even functions such that Ψn (t) > 0 for t > 0 and Ψn (|t|) ↓ as |t| ↑, |t|p
Ψn (|t|) ↑ and
where 0 < p ≤ 1. Then condition (1.12) implies (1.14). Corollary 1 Let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables with EXni = 0(1 ≤ i ≤ n, n ≥ 1). If max E|Xni |ν = O(nα ),
(1.15)
1≤i≤n
where
ν q
− α > max{ νr , α}, 0 < r ≤ 2, α > 0, ν ≥ max{r, 1}, then 1 n1/q
max |
1≤≤n
n
Xni | → 0 completely.
(1.16)
i=1
If we relax the conditions of Theorem 1 and Theorem 2, we can obtain the results of convergence in probability for arrays of rowwise NA random variables.. Theorem 4 Let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables, {an , n ≥ 1} a sequence of positive real numbers with an ↑ ∞, and {Ψn (t), n ≥ 1} a sequence of nonnegative even functions such that Ψn (t) > 0 for t > 0 and Ψn (|t|) Ψn (|t|) ↑ and ↓ as |t| ↑, |t| |t|p EXni = 0, 1 ≤ i ≤ n, n ≥ 1. 1) 1 < p ≤ 2. If
n
Ψi (Xni ) → 0 as n → ∞, Ψi (an )
(1.17)
n 1 Xni → 0 in probability. an i=1
(1.18)
E
i=1
then
2) p > 2. If (1.17) holds and n
i=1
E|Xni |r arn
→ 0 as n → ∞,
(1.19)
where 0 < r ≤ 2, then we have (1.18). Corollary 2 Let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables with EXni = 0 (1 ≤ i ≤ n, n ≥ 1). If max E|Xni |ν = O(nα ),
(1.20)
1≤i≤n
where α ≥ 0, ν ≥ max{r, 1}, 0 < r ≤ 2, then for any 0 < q < n 1
n1/q
i=1
ν α+max{ν/r,1} ,
Xni → 0 in probability.
we have (1.21)
No.2
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Theorem 5 Let {Xni , 1 ≤ n, i < ∞} be an array of any random variables, {an , n ≥ 1} a sequence of positive real numbers with an ↑ ∞, and {Ψn (t), n ≥ 1} a sequence of nonnegative even functions such that Ψn (t) > 0 for t > 0 and Ψn (|t|) ↑ and where 0 < p ≤ 1. If
Ψn (|t|) ↓ as |t| ↑, |t|p
n EΨi (Xni ) i=1
then
Ψi (ai )
→ 0 as n → ∞,
n 1 Xni → 0 in probability. an i=1
Remark Since an independent random variable sequence is a special NA sequence, Theorem 1 and Theorem 2 hold for arrays of rowwise independent random variables. So Theorem 1 and Theorem 2 are extensions and improvements of Theorem A and Theorem B.
2
Proof Proof of Theorem 1
Yni =
For any 1 ≤ i ≤ n, n ≥ 1, let
⎧ ⎪ ⎪ ⎨ an , ⎪ ⎪ ⎩
Xni > an ,
Xni , |Xni | ≤ an
Zni =
−an , Xni < −an ,
⎧ ⎪ ⎪ ⎨ Xni − an , Xni > an , ⎪ ⎪ ⎩
0,
|Xni | ≤ an ,
Xni + an , Xni < −an .
Clearly Xni = Yni + Zni , 1 ≤ i ≤ n, n ≥ 1. To prove (1.14) it suffices to show k 1 max Zni → 0 completely, an 1≤k≤n
(2.1)
i=1
k 1 max (Yni − EYni ) → 0 completely, an 1≤k≤n i=1 k 1 max EYni → 0 as n → ∞. an 1≤k≤n i=1
First, we prove (2.1). Since ∞
≤ ≤
Ψn (|t|) |t|
P
n=1 ∞ n n=1 i=1 ∞ n n=1 i=1
↑ as |t| ↑, then Ψn (|t|) ↑ as |t| ↑. So ∀ε > 0
k 1 max | Zni | > ε an 1≤k≤n i=1
P (Zni = 0) =
∞ n
P (|Xni | > an )
n=1 i=1 ∞ n
EΨi (|Xni| ) Ψi (Xni ) = < ∞. Ψi (an ) Ψi (an ) n=1 i=1
(2.2)
(2.3)
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Secondly, we prove (2.2). Note that {Yni − EYni , 1 ≤ i ≤ n, n ≥ 1} is an array of rowwise NA mean zero random variables by Lemma 1. From (1.2), (1.12), Markov inequality, Cr -inequality, and Jensen inequality for ∀ε > 0 ∞ n=1
≤
P
k 1 max | (Yni − EYni )| > ε an 1≤k≤n i=1
∞ k 1 1 E max | (Yni − EYni )|p p εp n=1 an 1≤k≤n i=1
≤c ≤c =c
∞ n E|Yni − EYni |p
n=1 i=1 ∞ n n=1 i=1 n ∞ n=1 i=1
apn
≤c
∞ n E|Yni |p
apn n=1 i=1 ∞ n EΨi (|Xni |)
EΨi (|Yni |) ≤c Ψi (an ) n=1 i=1
Ψi (an )
EΨi (Xni ) < ∞. Ψi (an )
Finally, we prove (2.3). For 1 ≤ i ≤ n, n ≥ 1, since EXni = 0, then EYni = −EZni . If Xni > an , 0 < Zni = Xni − an < Xni . If Xni < −ani , Xni < Zni = Xni + an ≤ 0. So |Zni | ≤ |Xni |. Consequently k k 1 1 max | EYni | = max | EZni | an 1≤k≤n i=1 an 1≤k≤n i=1
≤ ≤
n E|Zni | i=1 n i=1
≤
an
≤
n E|Xni |I(|Xni | > an )
EΨi (|Xni |)I(|Xni | > an ) Ψi (an )
n EΨi (Xni ) i=1
an
i=1
Ψi (an )
→ 0 as n → ∞.
The proof is complete. Proof of Theorem 2 Following the notations and the methods of the proof in Theorem 1, (2.1) and (2.3) hold. So we need only to show (2.2). Let ν = max{p, 2s}. By Markov inequality, (1.3), and Cr -inequality, it follows that ∞ n=1
P
k 1 max | (Yni − EYni )| > ε an 1≤k≤n i=1
∞ k 1 1 ≤ ν E max | (Yni − EYni )|ν ε n=1 aνn 1≤k≤n i=1 n ∞ n 1 ν 2 ν/2 ≤c E|Yni − EYni | + ( E(Yni − EYni ) ) aν n=1 n i=1 i=1 n ∞ n 1 ν 2 ν/2 ≤c E|Yni | + ( E|Yni | ) . aν n=1 n i=1 i=1
(2.4)
No.2
Gan & Chen: LIMITING BEHAVIOR OF MAXIMUM PARTIAL SUMS
(|t|) i (|t|) Since ν ≥ p, Ψ|t| ↓, Ψi|t| ↑, then p
Ψi (|t|) |t|ν
289
↓ and Ψi (|t|) ↑ as |t| ↑, therefore
n ∞ ∞ ∞ n n 1 EΨi (|Yni |) EΨi (|Xni |) ν ≤ E|Y | ≤ ni aν Ψi (an ) Ψi (an ) n=1 n i=1 n=1 i=1 n=1 i=1
=
∞ n EΨi (Xni )
Ψi (an )
n=1 i=1
< ∞.
(2.5)
since 0 < r ≤ 2, s > 0, ν ≥ 2s, then ν n 2s ∞ n ∞ E|Yni |2 1 ( E|Yni |2 )ν/2 = )s ( ν 2 a a n n n=1 n=1 i=1 i=1 ∞ n ν ∞ n ν EY 2 2s E|Yni |r 2s ni ≤ ( ) ≤ ( ) 2 a arn n n=1 i=1 n=1 i=1 ∞ n ν E|Xni |r 2s ≤ ( ) < ∞. arn n=1 i=1
(2.6)
From (2.4), (2.5), and (2.6) we obtain (2.2). The proof is complete. Proof of Theorem 3 Let Yni = Xni I(|Xni | ≤ an ), Zni = Xni I(|Xni | > an ), 1 ≤ i ≤ n, n ≥ 1. Obviously k k k 1 1 1 max | Xni | ≤ max | Yni | + max | Zni |. an 1≤k≤n i=1 an 1≤k≤n i=1 an 1≤k≤n i=1
To prove Theorem 3 we need only to show
and
k 1 max | Yni | → 0 completely, an 1≤k≤n i=1
(2.7)
k 1 max | Zni | → 0 completely. an 1≤k≤n i=1
(2.8)
For any ∀ε > 0, by Markov inequality and Cr -inequality ∞ n=1
≤c
P
k 1 max | Yni | > ε an 1≤k≤n i=1
∞ k ∞ n 1 E|Yni |p p E max | Y | ≤ c ni apn 1≤k≤n apn n=1 n=1 i=1
≤c =c
∞ n EΨi (|Yni |)
n=1 i=1 ∞ n n=1 i=1
ψi (an )
≤c
i=1
∞ n EΨi (|Xni |)
n=1 i=1
ψi (an )
EΨi (Xni ) < ∞. ψi (an )
We can prove (2.8) similarly as (2.1) in the proof of Theorem 1. The proof is complete.
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Proof of Corollary 1
Vol.27 Ser.B
Let an = n1/q , Ψn (t) = |t|ν , n ≥ 1, p = ν + 2. Since ν ≥ 1, then
Ψ(|t|) Ψ(|t|) 1 = |t|ν−1 ↑ and = 2 ↓ as |t| ↑ . |t| |t|p |t| From (1.15) and
ν q
− α > max{ νr , 2} it follows that
∞ n Ψi (Xni ) n=1 i=1
Since
ν q
Ψi (an )
− α > max{ νr , 2}, then
r q
=
∞ n E|Xni |ν n=1 i=1
−
αr ν
nν/q
≤c
∞
1 < ∞. ν/q−α−1 n n=1
− 1 > 0. Take s > 0 such that s( rq −
αr ν
(2.9) − 1) > 1, then
n s n s ∞ ∞ ∞ E|Xni |r (nα )r/ν 1 ≤ c = c < ∞. r r/q s(r/q−αr/ν−1) an n n n=1 i=1 n=1 i=1 n=1 It follows from Theorem 2 that 1 n1/q
max |
1≤k≤n
k
Xni | → 0 completely.
i=1
The proof of Theorem 4 is similar to that of Theorem 1 and Theorem 2. The proof of Corollary 2 is similar to the proof of Corollary 1. The proof of Theorem 5 is similar to the proof of Theorem 3. Therefore these proofs are all omitted. References 1 Hu Tienchung, Taylor R L. On the strong law for arrays and for the bootstrap mean and variance. Internat J Math & Math Sci, 1997, 20(2): 375–382 2 Hu Tienchung, Chang Henchao. Strong laws of larege numbers for arrays of random elements. Soochow J of Mathemativs, 1994, 20(4): 587–594 3 Joag-Dev K, Proschan F. Negative association of random variables with applications. Ann Statist, 1983, 11: 286–295 4 Su Chun, Tiang Tao, Tang Qihe, Liang Hanying. On the safety of NA dependence structure. Chinese Journal of Applied Probability and Statistics, 2002, 16(4): 400–404 5 Newman C M. Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong Y L, ed. Inequalities in Statistics and Probability. Institute of Mathematical Statistics, Hayward, CA, 1984. 127–140 6 Matula P. A note on the almost sure convergence of sums of negatively dependent random variables, Statist. Probab Lett, 1992, 15(3): 209–213 7 Shao Qiman, Su Chun. The law of the iterated logarithm for negatively associated random variables. Stochastic Processes and Their Applications, 1999, 83: 139–148 8 Shao Qiman. A comparison theorem on moment inequalities between negatively associated and independent random variables. J of Theoretical Probability, 2000, 13(2): 343–356 9 Liu Jingjun, Gan Shixin, Chen Pingyan. The H` ajeck–R` enyi inequality for NA random variables and its application. Statis Probab Lett, 1999, 43: 99–105 10 Barbour A D, Holst L, Janson S. Poisson Approximation. Oxford: Oxford University Press, 1992 11 Hsu P L, Robbins H. Complete convergence and the law of large numbers. Proc Nat Acad Sci, 1947, 33: 25–31
ON THE LIMITING BEHAVIOR OF THE MAXIMUM PARTIAL SUMS FOR ARRAYS OF ROWWISE NA RANDOM VARIABLES∗ Gan Shixin (
)
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China E-mail: [email protected]
Chen Pingyan (
)
Department of Mathematics, Jinan University, Guangzhou 510630, China
Abstract Let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables and {an , n ≥ 1} a sequence of constants with 0 < an ↑ ∞. The limiting behavior of maximum partial sums
k 1 max | Xni | an 1≤k≤n i=1
is investigated and some new results are obtained. The
results extend and improve the corresponding theorems of rowwise independent random variable arrays by Hu and Taylor [1] and Hu and Chang [2]. Key words NA random variable, maximum partial sum, complete convergence, convergence in probability 2000 MR Subject Classification
1
60F15, 60G50
Introduction and Main Results
Let (Ω, F , P ) be a probability space and {Xn , n ≥ 1} a sequence of random variables defined on (Ω, F , P ). We start with definitions. A finite family of random variables {Xi , 1 ≤ i ≤ n} is said to be negatively associated(abbreviated to NA) if for any disjoint subsets A and B of {1, 2, · · · , n} and any real coordinatewise nondecreasing functions f on RA and g on RB , Cov (f (Xi , i ∈ A), g(Xj , j ∈ B)) ≤ 0,
(1.1)
whenever the covariance exists. An infinite family of random variables {Xi , 1 ≤ i < ∞} is NA if every finite subfamily is NA. This concept was introduced by Joag–Dev and Proschan [3]. They also pointed out and proved in their paper that a number of well-known multivariate distributions process the NA property. Now people know NA random variables have wide applications in reliability theory and multivariate statistical analysis. Recently Su et al. [4] showed that NA structure plays an important role in risk management. Because of these reasons ∗ Received
February 10, 2004
284
ACTA MATHEMATICA SCIENTIA
Vol.27 Ser.B
the notions of NA random variables have received more and more attention in recent years. A great number of papers for NA random variables are now in literature. We refer to Joag–Dev and Proschan [3] for fundamental properties, Newman [5] for the central limit theorem, Matula [6] for the three series theorem, Shao and Su [7] for the law of iterated logarithm, Shao [8] for moment inequalities, Liu et al. [9] for the H` ajeck–R`enyi inequality and Barbour et al. [10] for Poison approximation. Recently Shao [8] showed the following important moment inequalities: let p > 1, {Xi , 1 ≤ i ≤ n} be a sequence of NA mean zero random variables with E|Xi |p < ∞ for every 1 ≤ i ≤ n. Then n k Xi |p ≤ 23−p E|Xi |p for 1 < p ≤ 2, (1.2) E max | 1≤k≤n
and E max | 1≤k≤n
k
i=1
i=1
p
p
Xi | ≤ 2(15p/ ln p)
n n p EXi2 ) 2 + E|Xi |p (
i=1
i=1
for p > 2.
(1.3)
i=1
For a triangular array of rowwise independent random variables {Xni , 1 ≤ i ≤ n, n ≥ 1} , a sequence of positive real numbers {an , n ≥ 1} with an ↑ ∞, and Ψ(t) a positive, even function such that ψ(|t|) Ψ(|t|) ↑ and ↓ as |t| ↑, (1.4) p |t| |t|p+1 for some nonnegative integer p. Conditions are given as EXni = 0, 1 ≤ i ≤ n, n ≥ 1, ∞ n
E
n=1 i=1
n ∞ n=1
i=1
Ψ(Xni ) < ∞, Ψ(an )
E
Xni an
(1.5) (1.6)
2 2k < ∞,
(1.7)
where k is a positive integer. In 1997 Hu and Taylor [1] proved the following theorems: Theorem A Let {Xni , 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise independent random variables and let Ψ(t) satisfy (1.4) for some integer p > 2. Then conditions (1.5), (1.6), and (1.7) imply n 1 Xni → 0 a.s.. (1.8) an i=1 Theorem B Let {Xni , 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise independent random variables and let Ψ(t) satisfy (1.4) for p = 1. Then conditions (1.5), (1.6) imply (1.8). Theorem C Let {Xni , 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise independent random variables and let Ψ(t) satisfy (1.4) for p = 0. Then condition (1.6) implies (1.8). The main purpose of this article is to utilize Shao’s inequalities to extend Theorem A and Theorem B to the case of NA random variables and Theorem C to the case of any random variables. It is worth to point out that our conclusions are much stronger and conditions are more general and much weaker. To prove our main results we need the following lemma:
No.2
Gan & Chen: LIMITING BEHAVIOR OF MAXIMUM PARTIAL SUMS
285
Lemma 1 Nondecreasing functions defined on disjoint subsets of a set of negatively associated random variables are negatively associated (see [3]). A sequence of random variables {Un , n ≥ 1} is said to converge completely to a constant a if for any ε > 0, ∞ P (|Un − a| > ε) < ∞. (1.9) n=1
In this case we write Un → a completely. This notion was given firstly by Hsu and Robbins [11]. Obviously the complete convergence implies the almost sure convergence. In this article, c always stands for a positive constant, which may differ from one place to another. Now, let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables and {an , n ≥ 1} a sequence of positive real numbers with an ↑ ∞. {Ψn (t), n ≥ 1} is a sequence of nonnegative even functions such that Ψn (t) > 0 as t > 0 and Ψn (|t|) Ψn (|t|) ↑ and ↓ as |t| ↑ . |t| |t|p
(1.10)
Conditions (1.11)–(1.13) are given as EXni = 0, 1 ≤ i ≤ n, n ≥ 1, ∞ n n=1 i=1
E
Ψi (Xni ) < ∞, Ψi (an )
n s ∞ E|Xni |r < ∞, arn n=1 i=1
(1.11) (1.12)
(1.13)
where 0 < r ≤ 2, s > 0. Theorem 1 Let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables, {an , n ≥ 1} a sequence of positive real numbers with an ↑ ∞ and {Ψn (t), n ≥ 1} a sequence of nonnegative even functions such that Ψn (t) > 0 for t > 0 and Ψn (|t|) Ψn (|t|) ↑ and ↓ as |t| ↑, |t| |t|p where 1 < p ≤ 2. Then Conditions (1.11) and (1.12) imply k 1 max | Xni | → 0 completely. an 1≤k≤n i=1
(1.14)
Theorem 2 Let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables, {an , n ≥ 1} a sequence of positive real numbers with an ↑ ∞ and {Ψn (t), n ≥ 1} a sequence of nonnegative even functions such that Ψn (t) > 0 for t > 0 and Ψn (|t|) Ψn (|t|) ↑ and ↓ as |t| ↑, |t| |t|p where p > 2. Then conditions (1.11), (1.12), and (1.13) imply (1.14).
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Theorem 3 Let {Xni , 1 ≤ n, i < ∞} be an array of any random variables, {an , n ≥ 1} a sequence of positive real numbers with an ↑ ∞ and {Ψn (t), n ≥ 1} a sequence of nonnegative even functions such that Ψn (t) > 0 for t > 0 and Ψn (|t|) ↓ as |t| ↑, |t|p
Ψn (|t|) ↑ and
where 0 < p ≤ 1. Then condition (1.12) implies (1.14). Corollary 1 Let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables with EXni = 0(1 ≤ i ≤ n, n ≥ 1). If max E|Xni |ν = O(nα ),
(1.15)
1≤i≤n
where
ν q
− α > max{ νr , α}, 0 < r ≤ 2, α > 0, ν ≥ max{r, 1}, then 1 n1/q
max |
1≤≤n
n
Xni | → 0 completely.
(1.16)
i=1
If we relax the conditions of Theorem 1 and Theorem 2, we can obtain the results of convergence in probability for arrays of rowwise NA random variables.. Theorem 4 Let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables, {an , n ≥ 1} a sequence of positive real numbers with an ↑ ∞, and {Ψn (t), n ≥ 1} a sequence of nonnegative even functions such that Ψn (t) > 0 for t > 0 and Ψn (|t|) Ψn (|t|) ↑ and ↓ as |t| ↑, |t| |t|p EXni = 0, 1 ≤ i ≤ n, n ≥ 1. 1) 1 < p ≤ 2. If
n
Ψi (Xni ) → 0 as n → ∞, Ψi (an )
(1.17)
n 1 Xni → 0 in probability. an i=1
(1.18)
E
i=1
then
2) p > 2. If (1.17) holds and n
i=1
E|Xni |r arn
→ 0 as n → ∞,
(1.19)
where 0 < r ≤ 2, then we have (1.18). Corollary 2 Let {Xni , 1 ≤ n, i < ∞} be an array of rowwise NA random variables with EXni = 0 (1 ≤ i ≤ n, n ≥ 1). If max E|Xni |ν = O(nα ),
(1.20)
1≤i≤n
where α ≥ 0, ν ≥ max{r, 1}, 0 < r ≤ 2, then for any 0 < q < n 1
n1/q
i=1
ν α+max{ν/r,1} ,
Xni → 0 in probability.
we have (1.21)
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287
Theorem 5 Let {Xni , 1 ≤ n, i < ∞} be an array of any random variables, {an , n ≥ 1} a sequence of positive real numbers with an ↑ ∞, and {Ψn (t), n ≥ 1} a sequence of nonnegative even functions such that Ψn (t) > 0 for t > 0 and Ψn (|t|) ↑ and where 0 < p ≤ 1. If
Ψn (|t|) ↓ as |t| ↑, |t|p
n EΨi (Xni ) i=1
then
Ψi (ai )
→ 0 as n → ∞,
n 1 Xni → 0 in probability. an i=1
Remark Since an independent random variable sequence is a special NA sequence, Theorem 1 and Theorem 2 hold for arrays of rowwise independent random variables. So Theorem 1 and Theorem 2 are extensions and improvements of Theorem A and Theorem B.
2
Proof Proof of Theorem 1
Yni =
For any 1 ≤ i ≤ n, n ≥ 1, let
⎧ ⎪ ⎪ ⎨ an , ⎪ ⎪ ⎩
Xni > an ,
Xni , |Xni | ≤ an
Zni =
−an , Xni < −an ,
⎧ ⎪ ⎪ ⎨ Xni − an , Xni > an , ⎪ ⎪ ⎩
0,
|Xni | ≤ an ,
Xni + an , Xni < −an .
Clearly Xni = Yni + Zni , 1 ≤ i ≤ n, n ≥ 1. To prove (1.14) it suffices to show k 1 max Zni → 0 completely, an 1≤k≤n
(2.1)
i=1
k 1 max (Yni − EYni ) → 0 completely, an 1≤k≤n i=1 k 1 max EYni → 0 as n → ∞. an 1≤k≤n i=1
First, we prove (2.1). Since ∞
≤ ≤
Ψn (|t|) |t|
P
n=1 ∞ n n=1 i=1 ∞ n n=1 i=1
↑ as |t| ↑, then Ψn (|t|) ↑ as |t| ↑. So ∀ε > 0
k 1 max | Zni | > ε an 1≤k≤n i=1
P (Zni = 0) =
∞ n
P (|Xni | > an )
n=1 i=1 ∞ n
EΨi (|Xni| ) Ψi (Xni ) = < ∞. Ψi (an ) Ψi (an ) n=1 i=1
(2.2)
(2.3)
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Secondly, we prove (2.2). Note that {Yni − EYni , 1 ≤ i ≤ n, n ≥ 1} is an array of rowwise NA mean zero random variables by Lemma 1. From (1.2), (1.12), Markov inequality, Cr -inequality, and Jensen inequality for ∀ε > 0 ∞ n=1
≤
P
k 1 max | (Yni − EYni )| > ε an 1≤k≤n i=1
∞ k 1 1 E max | (Yni − EYni )|p p εp n=1 an 1≤k≤n i=1
≤c ≤c =c
∞ n E|Yni − EYni |p
n=1 i=1 ∞ n n=1 i=1 n ∞ n=1 i=1
apn
≤c
∞ n E|Yni |p
apn n=1 i=1 ∞ n EΨi (|Xni |)
EΨi (|Yni |) ≤c Ψi (an ) n=1 i=1
Ψi (an )
EΨi (Xni ) < ∞. Ψi (an )
Finally, we prove (2.3). For 1 ≤ i ≤ n, n ≥ 1, since EXni = 0, then EYni = −EZni . If Xni > an , 0 < Zni = Xni − an < Xni . If Xni < −ani , Xni < Zni = Xni + an ≤ 0. So |Zni | ≤ |Xni |. Consequently k k 1 1 max | EYni | = max | EZni | an 1≤k≤n i=1 an 1≤k≤n i=1
≤ ≤
n E|Zni | i=1 n i=1
≤
an
≤
n E|Xni |I(|Xni | > an )
EΨi (|Xni |)I(|Xni | > an ) Ψi (an )
n EΨi (Xni ) i=1
an
i=1
Ψi (an )
→ 0 as n → ∞.
The proof is complete. Proof of Theorem 2 Following the notations and the methods of the proof in Theorem 1, (2.1) and (2.3) hold. So we need only to show (2.2). Let ν = max{p, 2s}. By Markov inequality, (1.3), and Cr -inequality, it follows that ∞ n=1
P
k 1 max | (Yni − EYni )| > ε an 1≤k≤n i=1
∞ k 1 1 ≤ ν E max | (Yni − EYni )|ν ε n=1 aνn 1≤k≤n i=1 n ∞ n 1 ν 2 ν/2 ≤c E|Yni − EYni | + ( E(Yni − EYni ) ) aν n=1 n i=1 i=1 n ∞ n 1 ν 2 ν/2 ≤c E|Yni | + ( E|Yni | ) . aν n=1 n i=1 i=1
(2.4)
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(|t|) i (|t|) Since ν ≥ p, Ψ|t| ↓, Ψi|t| ↑, then p
Ψi (|t|) |t|ν
289
↓ and Ψi (|t|) ↑ as |t| ↑, therefore
n ∞ ∞ ∞ n n 1 EΨi (|Yni |) EΨi (|Xni |) ν ≤ E|Y | ≤ ni aν Ψi (an ) Ψi (an ) n=1 n i=1 n=1 i=1 n=1 i=1
=
∞ n EΨi (Xni )
Ψi (an )
n=1 i=1
< ∞.
(2.5)
since 0 < r ≤ 2, s > 0, ν ≥ 2s, then ν n 2s ∞ n ∞ E|Yni |2 1 ( E|Yni |2 )ν/2 = )s ( ν 2 a a n n n=1 n=1 i=1 i=1 ∞ n ν ∞ n ν EY 2 2s E|Yni |r 2s ni ≤ ( ) ≤ ( ) 2 a arn n n=1 i=1 n=1 i=1 ∞ n ν E|Xni |r 2s ≤ ( ) < ∞. arn n=1 i=1
(2.6)
From (2.4), (2.5), and (2.6) we obtain (2.2). The proof is complete. Proof of Theorem 3 Let Yni = Xni I(|Xni | ≤ an ), Zni = Xni I(|Xni | > an ), 1 ≤ i ≤ n, n ≥ 1. Obviously k k k 1 1 1 max | Xni | ≤ max | Yni | + max | Zni |. an 1≤k≤n i=1 an 1≤k≤n i=1 an 1≤k≤n i=1
To prove Theorem 3 we need only to show
and
k 1 max | Yni | → 0 completely, an 1≤k≤n i=1
(2.7)
k 1 max | Zni | → 0 completely. an 1≤k≤n i=1
(2.8)
For any ∀ε > 0, by Markov inequality and Cr -inequality ∞ n=1
≤c
P
k 1 max | Yni | > ε an 1≤k≤n i=1
∞ k ∞ n 1 E|Yni |p p E max | Y | ≤ c ni apn 1≤k≤n apn n=1 n=1 i=1
≤c =c
∞ n EΨi (|Yni |)
n=1 i=1 ∞ n n=1 i=1
ψi (an )
≤c
i=1
∞ n EΨi (|Xni |)
n=1 i=1
ψi (an )
EΨi (Xni ) < ∞. ψi (an )
We can prove (2.8) similarly as (2.1) in the proof of Theorem 1. The proof is complete.
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Vol.27 Ser.B
Let an = n1/q , Ψn (t) = |t|ν , n ≥ 1, p = ν + 2. Since ν ≥ 1, then
Ψ(|t|) Ψ(|t|) 1 = |t|ν−1 ↑ and = 2 ↓ as |t| ↑ . |t| |t|p |t| From (1.15) and
ν q
− α > max{ νr , 2} it follows that
∞ n Ψi (Xni ) n=1 i=1
Since
ν q
Ψi (an )
− α > max{ νr , 2}, then
r q
=
∞ n E|Xni |ν n=1 i=1
−
αr ν
nν/q
≤c
∞
1 < ∞. ν/q−α−1 n n=1
− 1 > 0. Take s > 0 such that s( rq −
αr ν
(2.9) − 1) > 1, then
n s n s ∞ ∞ ∞ E|Xni |r (nα )r/ν 1 ≤ c = c < ∞. r r/q s(r/q−αr/ν−1) an n n n=1 i=1 n=1 i=1 n=1 It follows from Theorem 2 that 1 n1/q
max |
1≤k≤n
k
Xni | → 0 completely.
i=1
The proof of Theorem 4 is similar to that of Theorem 1 and Theorem 2. The proof of Corollary 2 is similar to the proof of Corollary 1. The proof of Theorem 5 is similar to the proof of Theorem 3. Therefore these proofs are all omitted. References 1 Hu Tienchung, Taylor R L. On the strong law for arrays and for the bootstrap mean and variance. Internat J Math & Math Sci, 1997, 20(2): 375–382 2 Hu Tienchung, Chang Henchao. Strong laws of larege numbers for arrays of random elements. Soochow J of Mathemativs, 1994, 20(4): 587–594 3 Joag-Dev K, Proschan F. Negative association of random variables with applications. Ann Statist, 1983, 11: 286–295 4 Su Chun, Tiang Tao, Tang Qihe, Liang Hanying. On the safety of NA dependence structure. Chinese Journal of Applied Probability and Statistics, 2002, 16(4): 400–404 5 Newman C M. Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong Y L, ed. Inequalities in Statistics and Probability. Institute of Mathematical Statistics, Hayward, CA, 1984. 127–140 6 Matula P. A note on the almost sure convergence of sums of negatively dependent random variables, Statist. Probab Lett, 1992, 15(3): 209–213 7 Shao Qiman, Su Chun. The law of the iterated logarithm for negatively associated random variables. Stochastic Processes and Their Applications, 1999, 83: 139–148 8 Shao Qiman. A comparison theorem on moment inequalities between negatively associated and independent random variables. J of Theoretical Probability, 2000, 13(2): 343–356 9 Liu Jingjun, Gan Shixin, Chen Pingyan. The H` ajeck–R` enyi inequality for NA random variables and its application. Statis Probab Lett, 1999, 43: 99–105 10 Barbour A D, Holst L, Janson S. Poisson Approximation. Oxford: Oxford University Press, 1992 11 Hsu P L, Robbins H. Complete convergence and the law of large numbers. Proc Nat Acad Sci, 1947, 33: 25–31