Т ЕО Р ИЯ В ЕР О ЯТ Н ОС Т ЕЙТ ом 62 И Е Е П РИ М ЕН Е НИ Я В ып у с к 3

2017

c⃝ 2017 г. WANG X. J.∗, HU S. H.∗, VOLODIN A. I.∗∗

MOMENT INEQUALITIES FOR m-NOD RANDOMVARIABLES AND THEIR APPLICATIONS1)

Вводится понятие m-отрицательно ортант зависимых (сокращен-но m-NOD) случайных величин и для них устанавливаются момент-ные неравенства, такие как неравенство Марцинкевича–Зигмунда иРозенталя. Как одно из применений моментных неравенств изуча-ются Lr- и почти наверное сходимости для m-NOD случайных вели-чин при определенных условиях на равномерную интегрируемость.С другой стороны, устанавливается асимптотическое разложение об-ратных моментов для неотрицательных m-NOD случайных величинс конечными начальными моментами. Результаты статьи обобщаютили улучшают некоторые известные результаты для независимых инекоторых классов зависимых последовательностей.

Ключевые слова и фразы: m-отрицательно ортант зависимые слу-чайные величины; Lr-сходимость; обратные моменты; неравенстваМарцинкевича–Зигмунда; неравенства Розенталя.

DOI: https://doi.org/10.4213/tvp5123

1. Introduction. It is well known that the Marcinkiewicz–Zygmundtype inequality and Rosenthal type inequality play important role in prob-ability limit theory and mathematical statistics, especially in establishingstrong convergence, complete convergence, weak convergence, consistencyand asymptotic normality in many stochastic models. There are many se-quences of random variables satisfying the Marcinkiewicz–Zygmund type in-equality or Rosenthal type inequality under some suitable conditions, suchas independent sequence, ϕ-mixing sequence with the mixing coefficients sat-isfying certain conditions (see [36]), ρ-mixing sequence with the mixing co-efficients satisfying certain conditions (see [20]), ρ-mixing sequence (see [32]

∗School of Mathematical Sciences, Anhui University, China; e-mail: wxjahdx@126.com∗∗Department of Mathematics and Statistics, University of Regina, Regina, Canada.1)Supported by the National Natural Science Foundation of China (11671012, 11501004,

11501005), the Natural Science Foundation of Anhui Province (1508085J06), and the KeyProjects for Academic Talent of Anhui Province (gxbjZD2016005).

588 Wang X. J., Hu S. H., Volodin A. I.

or [40]), negatively associated sequence (NA, in short, see [21]), negativelyorthant dependent sequence (NOD, in short, see [2]), extended negativelydependent sequence (END, in short, see [22]), negatively superadditive de-pendent sequence (NSD, in short, see [11] or [39]), asymptotically almostnegatively associated sequence with the mixing coefficients satisfying certainconditions (AANA, in short, see [49]), ρ−-mixing sequence with the mixingcoefficients satisfying certain conditions (see [34]), and so on.

The main purpose of the paper is to introduce a new concept of depen-dent structure — m-negatively orthant dependence (m-NOD, in short) andestablish the Marcinkiewicz–Zygmund type inequality and Rosenthal typeinequality for m-NOD random variables. In addition, we will give someapplications of Marcinkiewicz–Zygmund type inequality and Rosenthal typeinequality to Lr convergence, strong law of large numbers and the asymptoticapproximation of inverse moments for nonnegative m-NOD random variableswith finite first moments.

Firstly, let us recall the definition of negatively orthant dependent randomvariables which was introduced by Joav-Dev and Proschan [14] as follows.

Definition 1.1. A finite collection of random variables X1, . . . , Xn is saidto be negatively orthant dependent (NOD, in short) if both

P(X1 > x1, . . . , Xn > xn) 6n∏

i=1

P(Xi > xi)

and

P(X1 6 x1, . . . , Xn 6 xn) 6n∏

i=1

P(Xi 6 xi)

hold for each n > 1 and all real numbers x1, . . . , xn. An infinite sequence{Xn, n > 1} is said to be NOD if every finite subcollection is NOD.

An array {Xni, i > 1, n > 1} of random variables is said to be rowwiseNOD if for every n > 1, {Xni, i > 1} is a sequence of NOD random variables.

The class of NOD random variables is a very general dependent structure,which includes independent random variables and NA random variables asspecial cases. For more details about the probability inequalities, momentinequalities, or probability limit theory and applications, one can refer to[14], [4], [33], [31], [16], [2], [41], [37], [38], [42], [50], [19], [25], [45], and etc.

Inspired by the definition of NOD random variables, we introduce theconcept of m-negatively orthant dependent random variables as follows.

Definition 1.2. Let m > 1 be a fixed integer. A sequence {Xn, n > 1}of random variables is said to be m-negatively orthant dependent (m-NOD,in short) if for any n > 2 and any i1, . . . , in such that |ik − ij | > m for all1 6 k = j 6 n, we have that Xi1 , . . . , Xin are negatively orthant dependent.

Moment inequalities for m-NOD random variables and their applications 589

An array {Xni, i > 1, n > 1} of random variables is said to be rowwisem-NOD if for every n > 1, {Xni, i > 1} is a sequence of m-NOD randomvariables.

When n = 2, m-NOD reduces to m-pairwise NOD which was introducedby Anh in [1], and carefully studied by Wu and Rosalsky in [46]. Whenm = 1, the concept m-NOD random variables reduces to the so-called NODrandom variables. Hence, the concept of m-NOD random variables is a nat-ural extension from NOD random variables. Joav-Dev and Proschan in [14]pointed out that NA implies NOD, but NOD does not implies NA. Hu andYang [12] or Hu [13] pointed out that NSD implies NOD. Hence, the class ofm-NOD random variables includes independent random variables, NA ran-dom variables, NSD random variables, NOD random variables, and m-NArandom variables (see [10]) as special cases. Studying the probability inequal-ities, moment inequalities, limiting behavior of m-NOD random variables andtheir applications in many stochastic models are of great interest.

The following lemmas for NOD random variables will be used in estab-lishing the Marcinkiewicz–Zygmund type inequality and Rosenthal type in-equality for m-NOD random variables.

Lemma 1.1 (cf. [4]). Let random variables X1, . . . , Xn be NOD, f1, . . . , fnbe all nondecreasing (or all nonincreasing) functions, then random variablesf1(X1), . . . , fn(Xn) are NOD.

Lemma 1.2 (cf. [2]). Let {Xn, n > 1} be a sequence of NOD randomvariables with EXn = 0 and E|Xn|p < ∞ for some p > 1 and every n > 1.Then there exist positive constants Cp and Dp depending only on p such thatfor every n > 1,

E

∣∣∣∣ n∑i=1

Xi

∣∣∣∣p 6 Cp

n∑i=1

E|Xi|p for 1 6 p 6 2 (1.1)

and

E

∣∣∣∣ n∑i=1

Xi

∣∣∣∣p 6 Dp

{ n∑i=1

E|Xi|p +( n∑

i=1

EX2i

)p/2}for p > 2. (1.2)

Throughout the paper, let C denote a positive constant not dependingon n, which may be different in various places; an = O(bn) stands for an 6Cbn, where {an, n > 1} and {bn, n > 1} are sequences of nonnegative realnumbers. Denote log x = lnmax(x, e), x+ = xI(x > 0), x− = −xI(x < 0).

This work is organized as follows: the Marcinkiewicz–Zygmund type in-equality and Rosenthal type inequality for m-NOD random variables are pro-vided in Section 2. Some results on Lr convergence and strong law of largenumbers for arrays of rowwise m-NOD random variables are established in

590 Wang X. J., Hu S. H., Volodin A. I.

Section 3. The asymptotic approximation of inverse moments for nonneg-ative m-NOD random variables with finite first moments is investigated inSection 4.

2. Marcinkiewicz–Zygmund type inequality and Rosenthaltype inequality for m-NOD random variables. In this section,we will establish the Marcinkiewicz–Zygmund type inequality and Rosenthaltype inequality for m-NOD random variables, which can be applied to provethe strong convergence, Lr convergence, weak convergence, complete conver-gence, consistency and asymptotic normality in many stochastic models, andso on. To prove the main results, we need the following lemma, which willbe used frequently throughout the paper.

Lemma 2.1. Let {Xn, n > 1} be a sequence of m-NOD random variables.If {fn( · ), n > 1} are all nondecreasing (or nonincreasing) functions, thenrandom variables {fn(Xn), n > 1} are m-NOD.

This lemma can be obtained easily by the definition of m-NOD randomvariables and Lemma 1.1. So the details are omitted.

With Lemma 1.2 and Lemma 2.1 accounted for, we can establish theMarcinkiewicz–Zygmund type inequality and Rosenthal type inequality form-NOD random variables as follows.

Theorem 2.1. Let {Xn, n > 1} be a sequence of m-NOD random vari-ables with EXn = 0 and E|Xn|p < ∞ for some p > 1 and every n > 1.Then there exist positive constants Cm,p and Dm,p depending only on m andp such that for every n > m,

E

∣∣∣∣ n∑i=1

Xi

∣∣∣∣p6Cm,p

n∑i=1

E|Xi|p, for 1 6 p 6 2,

Dm,p

[ n∑i=1

E|Xi|p+( n∑

i=1

EX2i

)p/2], for p > 2,

(2.1)

and

E

(max16j6n

∣∣∣∣ j∑i=1

Xi

∣∣∣∣p)6

Cm,p lnp n

n∑i=1

E|Xi|p, for 1 6 p 6 2,

Dm,p lnp n

[ n∑i=1

E|Xi|p+( n∑

i=1

EX2i

)p/2],

for p > 2.

(2.2)

Proof. From (2.1), we can see that (2.2) can be obtained immediately bya similar way of the process of Theorem 2.3.1 in [28]. So we only need toprove (2.1).

Moment inequalities for m-NOD random variables and their applications 591

For fixed n > m, let r = ⌈n/m⌉. Define

Yi =

{Xi, 1 6 i 6 n,

0, i > n.

Denote S′mr+j =

∑ri=0 Ymi+j for j = 1, . . . ,m. Noting that

∑ni=1Xi =∑m

j=1 S′mr+j , we have by Cr-inequality that

E

∣∣∣∣ n∑i=1

Xi

∣∣∣∣p = E

∣∣∣∣ m∑j=1

S′mr+j

∣∣∣∣p 6 mp−1m∑j=1

E|S′mr+j |p. (2.3)

By definition of m-NOD random variables, we see that Yj , Ym+j , . . . , Ymr+j

are NOD random variables for each j = 1, . . . ,m.For 1 6 p 6 2, we have by (1.1) and (2.3) that for any n > m,

E

∣∣∣∣ n∑i=1

Xi

∣∣∣∣p 6 mp−1Cp

m∑j=1

r∑i=0

E|Ymi+j |p

6 mpCp

n∑i=1

E|Xi|p.= Cm,p

n∑i=1

E|Xi|p. (2.4)

For p > 2, we have by (1.2) and (2.3) that for any n > m,

E

∣∣∣∣ n∑i=1

Xi

∣∣∣∣p 6 mp−1Dp

m∑j=1

[ r∑i=0

E|Ymi+j |p +( r∑

i=0

EY 2mi+j

)p/2]

6 mpDp

[ n∑i=1

E|Xi|p +( n∑

i=1

EX2i

)p/2].= Dm,p

[ n∑i=1

E|Xi|p +( n∑

i=1

EX2i

)p/2]. (2.5)

The desired result (2.1) follows by (2.4) and (2.5) immediately. This com-pletes the proof of the theorem.

Remark 2.1. Assume that (2.1) holds for any n > m and∑∞

i=1Xi con-verges almost surely. Then

E

∣∣∣∣ ∞∑i=1

Xi

∣∣∣∣p 6Cm,p

∞∑i=1

E|Xi|p, for 1 6 p 6 2,

Dm,p

[ ∞∑i=1

E|Xi|p +( ∞∑

i=1

EX2i

)p/2], for p > 2.

(2.6)

592 Wang X. J., Hu S. H., Volodin A. I.

In fact, it follows by Fatou’s lemma that

E

∣∣∣∣ ∞∑i=1

Xi

∣∣∣∣p = E

∣∣∣∣lim infn→∞

n∑i=1

Xi

∣∣∣∣p 6 E

(lim infn→∞

∣∣∣∣ n∑i=1

Xi

∣∣∣∣p)

6 lim infn→∞

E

∣∣∣∣ n∑i=1

Xi

∣∣∣∣p 6 lim supn→∞

E

∣∣∣∣ n∑i=1

Xi

∣∣∣∣p, (2.7)

which together with (2.1) yields (2.6).Remark 2.2. Let {an, n > 1} be a sequence of real numbers. Under the

conditions of Theorem 2.1, we have for n > m that

E

∣∣∣∣ n∑i=1

aiXi

∣∣∣∣p6

2p−1Cm,p

n∑i=1

E|aiXi|p, for 1 6 p 6 2,

2pDm,p

[ n∑i=1

E|aiXi|p+( n∑

i=1

Ea2iX2i

)p/2],

for p > 2.

(2.8)

Assume further that∑∞

i=1 aiXi converges almost surely, we have for n > m

that

E

∣∣∣∣ ∞∑i=1

aiXi

∣∣∣∣p6

2p−1Cm,p

∞∑i=1

E|aiXi|p, for 1 6 p 6 2,

2pDm,p

[ ∞∑i=1

E|aiXi|p+( ∞∑

i=1

Ea2iX2i

)p/2],

for p > 2.

(2.9)

Actually, for fixed n > m, {a+i Xi, 1 6 i 6 n} and {a−i Xi, 1 6 i 6 n} areboth m-NOD random variables from Lemma 2.1. Noting that ani = a+ni−a−ni,we have by Cr-inequality that

E

∣∣∣∣ n∑i=1

aiXi

∣∣∣∣p 6 2p−1E

∣∣∣∣ n∑i=1

a+i Xi

∣∣∣∣p + 2p−1E

∣∣∣∣ n∑i=1

a−i Xi

∣∣∣∣p. (2.10)

Note that |ai|p = (a+i )p + (a−i )

p, the desired result (2.8) follows by (2.1)and (2.10) immediately.

Similar to the proof of (2.7), we can get (2.9) by (2.8) immediately.3. Lr convergence and strong convergence for m-NOD ran-

dom variables. In the previous section, we established the Marcinkiewicz–Zygmund type inequality and Rosenthal type inequality for m-NOD randomvariables. As one application of the moment inequalities for m-NOD ran-dom variables, we will study the Lr convergence and strong convergence form-NOD random variables under some uniformly integrable conditions.

Moment inequalities for m-NOD random variables and their applications 593

In what follows, let {un, n > 1} and {vn, n > 1} be two sequences ofintegers (not necessary positive or finite) such that vn > un for all n > 1 andvn − un → ∞ as n → ∞. Let {kn, n > 1} be a sequence of positive numberssuch that kn → ∞ as n → ∞ and {h(n), n > 1} be an increasing sequenceof positive constants with h(n) ↑ ∞ as n ↑ ∞.

3.1. Lr convergence and weak law of large numbers. The notionof h-integrability for an array of random variables concerning an array ofconstant weights was introduced by Cabrera and Volodin [17] as follows.

Definition 3.1. Let {Xni, un 6 i 6 vn, n > 1} be an array of randomvariables and let {ani, un 6 i 6 vn, n > 1} be an array of constants with∑vn

i=un|ani| 6 C for all n ∈ N and some constant C > 0. The array

{Xni, un 6 i 6 vn, n > 1} is said to be h-integrable with respect to thearray of constants {ani} if

supn>1

vn∑i=un

|ani|E|Xni| < ∞ and limn→∞

vn∑i=un

|ani|E|Xni|I(|Xni| > h(n)) = 0.

The main idea of the notion of h-integrability with respect to the arrayof constants {ani} is to deal with weighted sums of random variables. Sunget al. [29] introduced a new concept of integrability which deals with usualnormed sums of random variables as follows.

Definition 3.2. Let {Xni, un 6 i 6 vn, n > 1} be an array of randomvariables and r > 0. The array {Xni, un 6 i 6 vn, n > 1} is said to beh-integrable with exponent r if

supn>1

1

kn

vn∑i=un

E|Xni|r < ∞ and limn→∞

1

kn

vn∑i=un

E|Xni|rI(|Xni|r > h(n)) = 0.

Under the conditions of h-integrability with exponent r and h-integrabilitywith respect to the array of constants {ani}, Sung et al. [29] obtained thefollowing Theorem A and Theorem B for arrays of rowwise NA random vari-ables, respectively.

Theorem A. Let 1 6 r < 2. Suppose that {Xni, un 6 i 6 vn, n > 1} isan array of rowwise NA random variables. Let {ani, un 6 i 6 vn, n > 1} bean array of constants. Assume that the following conditions hold:

(i) {|Xni|r} is h-integrable concerning the array {|ani|r}, i.e.,

supn>1

vn∑i=un

|ani|rE|Xni|r < ∞

and

limn→∞

vn∑i=un

|ani|rE|Xni|rI(|Xni|r > h(n)) = 0;

594 Wang X. J., Hu S. H., Volodin A. I.

(ii) h(n) supun6i6vn |ani| → 0 as n → ∞.Then

vn∑i=un

ani(Xni −EXni) → 0

in Lr and, hence, in probability as n → ∞.Theorem B. Suppose that {Xni, un 6 i 6 vn, n > 1} is an array of

rowwise NA h-integrable with exponent 1 6 r < 2 random variables, kn → ∞,h(n) ↑ ∞, and h(n)/kn → 0 as n → ∞. Then∑vn

i=un(Xni −EXni)

k1/rn

→ 0

in Lr and, hence, in probability as n → ∞.Inspired by the concept of h-integrability with exponent r, Wang and

Hu [35] introduced a new and weaker concept of uniform integrability asfollows.

Definition 3.3. Let {Xni, un 6 i 6 vn, n > 1} be an array of randomvariables and r > 0. The array {Xni, un 6 i 6 vn, n > 1} is said to beresidually h-integrable (R-h-integrable, in short) with exponent r if

supn>1

1

kn

vn∑i=un

E|Xni|r < ∞

and

limn→∞

1

kn

vn∑i=un

E(|Xni| − h1/r(n)

)rI(|Xni|r > h(n)) = 0.

Under the assumption of R-h-integrability with exponent r, Wang andHu [35] established some weak laws of large numbers for arrays of dependentrandom variables. Note that(

|Xni| − h1/r(n))rI(|Xni|r > h(n)) 6 |Xni|rI(|Xni|r > h(n)),

hence, the concept of R-h-integrability with exponent r is weaker than h-inte-grability with exponent r.

Just as h-integrability with exponent r, the main idea of the notion ofR-h-integrability with exponent r is used to deal with usual normed sums ofrandom variables. We now introduce a new and weaker concept of integra-bility which deals with weighted sums of random variables.

Definition 3.4. Let {Xni, un 6 i 6 vn, n > 1} be an array of randomvariables and let {ani, un 6 i 6 vn, n > 1} be an array of constants. Letr > 0. The array {Xni, un 6 i 6 vn, n > 1} is said to be R-h-integrable withexponent r concerning the array {ani, un 6 i 6 vn, n > 1} if

supn>1

vn∑i=un

|ani|rE|Xni|r < ∞

Moment inequalities for m-NOD random variables and their applications 595

and

limn→∞

vn∑i=un

|ani|rE(|Xni| − h1/r(n)

)rI(|Xni|r > h(n)) = 0.

When r = 1, the notion of R-h-integrability with exponent r concern-ing the array of constants {ani} reduces to the so-called R-h-integrabilityconcerning the array of constants {ani}. For more details about the Lr con-vergence for weighted sums of random variables based on R-h-integrability,one can refer to [48], [25], and so on.

The main purpose of this section is to generalize and improve the results ofTheorem A and Theorem B for arrays of rowwise NA random variables to thecase of arrays of rowwise m-NOD random variables under some weaker con-ditions. In addition, we will study the Lr convergence and weak law of largenumbers for a class of random variables under the condition of h-integrabilitywith exponent 1 6 r < 2, which generalize the corresponding ones of [44] and[30], and improve the corresponding one of [6]. The key techniques used hereare the Marcinkiewicz–Zygmund type inequality and the truncated method.

Our main results on Lr convergence and weak law of large numbers forarrays of rowwise m-NOD are as follows. The first one is based on thecondition of R-h-integrability with exponent 1 6 r < 2 concerning the arrayof constants {ani}.

Theorem 3.1. Let 1 6 r < 2. Let {Xni, un 6 i 6 vn, n > 1} be an arrayof rowwise m-NOD random variables and let {ani, un 6 i 6 vn, n > 1} bean array of constants. Assume that the following conditions hold:

(i) {Xni, un 6 i 6 vn, n > 1} is R-h-integrable with exponent r concerningthe array of constants {ani}, i.e.,

supn>1

vn∑i=un

|ani|rE|Xni|r < ∞

and

limn→∞

vn∑i=un

|ani|rE(|Xni| − h1/r(n)

)rI(|Xni|r > h(n)) = 0;

(ii) h(n) supun6i6vn |ani|r → 0 as n → ∞.

Thenvn∑

i=un

ani(Xni −EXni) → 0

in Lr and, hence, in probability as n → ∞.Proof. Since ani = a+ni − a−ni, without loss of generality, we assume that

ani > 0. For fixed n > 1, denote for un 6 i 6 vn that

Yni = −h1/r(n)I(Xni < −h1/r(n)

)+XniI

(|Xni| 6 h1/r(n)

)

596 Wang X. J., Hu S. H., Volodin A. I.

+ h1/r(n)I(Xni > h1/r(n)

),

Zni = Xni − Yni =(Xni + h1/r(n)

)I(Xni < −h1/r(n)

)+(Xni − h1/r(n)

)I(Xni > h1/r(n)

),

Sn =

vn∑i=un

ani(Yni −EYni), Tn =

vn∑i=un

ani(Zni −EZni).

Note thatvn∑

i=un

ani(Xni −EXni) = Sn + Tn, n > 1,

we have by Cr-inequality that

E

∣∣∣∣ vn∑i=un

ani(Xni −EXni)

∣∣∣∣r 6 CE|Sn|r + CE|Tn|r.

To prove∑vn

i=unani(Xni−EXni) → 0 in Lr, we only need to show E|Sn|r → 0

and E|Tn|r → 0 as n → ∞, where 1 6 r < 2.Firstly, we will show that E|Sn|r → 0 as n → ∞. Note that 1 6 r < 2, it

suffices to show ES2n → 0 as n → ∞.

For fixed n> 1, it follows by Lemma 2.1 that {ani(Yni−EYni), un6 i6 vn}are m-NOD random variables. Note that |Yni| = min{|Xni|, h1/r(n)}, wehave by Theorem 2.1 and Remark 2.1 that

ES2n = E

∣∣∣∣ vn∑i=un

ani(Yni −EYni)

∣∣∣∣2 6 C

vn∑i=un

a2niEY 2ni

6 Ch(2−r)/r(n) supn>1

|ani|2−rvn∑

i=un

|ani|rE|Yni|r

6 C[h(n) sup

un6i6vn

|ani|r](2−r)/r

vn∑i=un

|ani|rE|Xni|r → 0 as n → ∞,

which implies that ES2n → 0 as n → ∞ and, thus, E|Sn|r → 0 as n → ∞.

Further, we will show that E|Tn|r → 0 as n → ∞.For fixed n > 1, it follows by Lemma 2.1 again that {ani(Zni−EZni), un 6

i 6 vn} are m-NOD random variables. Note that

|Zni| =(|Xni| − h1/r(n)

)I(|Xni| > h1/r(n)

),

we have by Theorem 2.1 and Remark 2.1 again that

E|Tn|r = E

∣∣∣∣ vn∑i=un

ani(Zni −EZni)

∣∣∣∣r 6 C

vn∑i=un

|ani|rE|Zni|r

= C

vn∑i=un

|ani|rE(|Xni| − h1/r(n)

)rI(|Xni|r > h(n)

)→ 0 as n → ∞,

Moment inequalities for m-NOD random variables and their applications 597

which implies that E|Tn|r → 0 as n → ∞. This completes the proof of thetheorem.

If we take ani = k−1/rn for un 6 i 6 vn and n > 1, then we can get

the following result on Lr convergence and weak law of large numbers forarrays of rowwise m-NOD R-h-integrable with exponent 1 6 r < 2 randomvariables.

Corollary 3.1. Let {Xni, un 6 i 6 vn, n > 1} be an array of rowwisem-NOD R-h-integrable random variables with exponent 1 6 r < 2, kn → ∞,h(n) ↑ ∞, and h(n)/kn → 0 as n → ∞. Then

1

k1/rn

vn∑i=un

(Xni −EXni) → 0

in Lr and, hence, in probability as n → ∞.Remark 3.1. Note that(

|Xni| − h1/r(n))rI(|Xni|r > h(n)) 6 |Xni|rI(|Xni|r > h(n)),

and h(n) supun6i6vn |ani| → 0 implies h(n) supun6i6vn |ani|r → 0 (here, 1 6

r < 2 and h(n) ↑ ∞ as n → ∞), which imply that the conditions of Theo-rem 3.1 are weaker than those of Theorem A. Hence, the result of Theorem 3.1generalizes and improves the corresponding one of Theorem A.

Remark 3.2. Since the concept of R-h-integrability with exponent r isweaker than h-integrability with exponent r and m-NOD is weaker thanNA, the result of Corollary 3.1 generalizes and improves the correspondingone of Theorem B.

Further, we will establish the Lr-convergence and weak law of large num-bers for a class of random variables satisfying the Marcinkiewicz–Zygmundinequality with exponent 2, which includes m-NOD as a special case. Themain ideas are inspired by [6] and [30].

We say that a sequence {Xn, n > 1} of random variables satisfies theMarcinkiewicz–Zygmund inequality with exponent 2, if for all n > 1,

E

∣∣∣∣ n∑i=1

Xi

∣∣∣∣2 6 C

n∑i=1

E|Xi|2,

where C is a positive constant not depending on n.We say that an array {Xni, un 6 i 6 vn, n > 1} of random variables

satisfies the Marcinkiewicz–Zygmund inequality with exponent 2, if for alln > 1,

E

∣∣∣∣ vn∑i=un

Xni

∣∣∣∣2 6 C

vn∑i=un

E|Xni|2,

where C is a positive constant not depending on n.

598 Wang X. J., Hu S. H., Volodin A. I.

Remark 3.3. There are many sequences of mean zero random variables sat-isfying the Marcinkiewicz–Zygmund inequality with exponent 2, such as in-dependent sequence, martingale difference sequence, ϕ-mixing sequence withthe mixing coefficients satisfying certain conditions (see [36]), ρ-mixing se-quence with the mixing coefficients satisfying certain conditions (see [20]),ρ-mixing sequence (see [32]), NA sequence (see [21]), NOD sequence (see [2]),END sequence (see [22]), NSD sequence (see [11] or [39]), AANA sequencewith the mixing coefficients satisfying certain conditions (see [49]), ρ−-mixingsequence with the mixing coefficients satisfying certain conditions (see [34]),pairwise negatively quadrant dependent sequence (PNQD, in short, see [17]),m-NOD sequence (see Theorem 2.1 in the paper), linearly negative quadrantdependent sequence (LNQD, in short, [51]), and so on.

Our main result on Lr convergence and weak law of large numbers for aclass of random variables satisfying the Marcinkiewicz–Zygmund inequalitywith exponent 2 is as follows.

Theorem 3.2. Let 1 6 r < 2. Let {Xni, un 6 i 6 vn, n > 1} be anarray of random variables and let {ani, un 6 i 6 vn, n > 1} be an array ofconstants. Assume that the following conditions hold:

(i) supn>1

∑vni=un

|ani|rE|Xni|r < ∞,(ii) for any ε > 0,

limn→∞

vn∑i=un

|ani|rE|Xni|rI(|Xni|r > ε) = 0,

(iii) for any t > 0, the array {Yni −EYni, un 6 i 6 vn, n > 1} satisfies theMarcinkiewicz–Zygmund inequality with exponent 2, where

Yni = aniXniI(|aniXni| 6 t1/r

)or

Yni = −t1/rI(aniXni < −t1/r

)+ aniXniI

(|aniXni| 6 t1/r

)+ t1/rI

(aniXni > t1/r

).

Thenvn∑

i=un

ani(Xni −EXni) → 0

in Lr and, hence, in probability as n → ∞.Proof. The proof is similar to that of [30]. So the details are omitted.With Theorem 3.2 in hand and similar to the proof of Corollary 2.1 in [30],

we can get the following corollary.Corollary 3.2. Let {ani, un 6 i 6 vn, n > 1} be an array of constants

satisfying kn.= 1/ supun6i6vn |ani|

r → ∞, 0 < h(n) ↑ ∞, and h(n)/kn → 0

Moment inequalities for m-NOD random variables and their applications 599

as n → ∞. Let {Xni, un 6 i 6 vn, n > 1} be an array of h-integrable randomvariables with exponent 1 6 r < 2. Assume further that the condition (iii)in Theorem 3.2 holds. Then

vn∑i=un

ani(Xni −EXni) → 0

in Lr and, hence, in probability as n → ∞.Taking ani = k

−1/rn for un 6 i 6 vn and n > 1 in Corollary 3.2, we can

get the following corollary immediately.Corollary 3.3. Let {Xni, un 6 i 6 vn, n > 1} be an array of h-integrable

with exponent 1 6 r < 2 random variables, kn → ∞, 0 < h(n) ↑ ∞,and h(n)/kn → 0 as n → ∞. Assume further that the condition (iii) inTheorem 3.2 holds, where ani = k

−1/rn for un 6 i 6 vn and n > 1. Then∑vn

i=un(Xni −EXni)

k1/rn

→ 0

in Lr and, hence, in probability as n → ∞.Remark 3.4. We have pointed out that PNQD sequence satisfies the

Marcinkiewicz–Zygmund inequality with exponent 2 in Remark 3.3. Hence,the results of Theorem 3.2 and Corollaries 3.2 and 3.3 in the paper generalizethe corresponding ones of Theorem 2.1 and Corollaries 2.1 and 2.2 for PNQDrandom variables in [30], respectively. In addition, note that LNQD impliesPNQD (see [30]), hence, our results of Corollary 3.2 and Corollary 3.3 gen-eralize the corresponding ones of Theorem 3.1 and Corollary 3.1 for LNQDrandom variables in [44], respectively.

Remark 3.5. Under the conditions of Corollary 3.2, Chen et al. in [6]established the L1 convergence and weak law of large numbers for arraysof rowwise h-integrable with exponent r = 1 random variables satisfyingthe Marcinkiewicz–Zygmund inequality with exponent 2. Here, we estab-lished the Lr convergence and weak law of large numbers for arrays ofrowwise h-integrable with exponent 1 6 r < 2 random variables satis-fying the Marcinkiewicz–Zygmund inequality with exponent 2. In addi-tion, the condition «kn

.= 1/ supun6i6vn |ani|

r → ∞, 0 < h(n) ↑ ∞ andh(n)/kn → 0 as n → ∞» in Corollary 3.2 in the paper is weaker than«kn

.= 1/ supun6i6vn |ani| → ∞, 0 < h(n) ↑ ∞ and h(n)/kn → 0 as n → ∞»

in Theorem 1 in [6]. Hence, our results of Theorem 3.2 and Corollary 3.2generalize and improve the corresponding one of Theorem 1 in [6].

3.2. Strong convergence. In Section 3.1, we studied the Lr conver-gence and weak law of large numbers for arrays of rowwise m-NOD randomvariables under some uniformly integrable conditions. In order to establishthe strong version of Theorem 3.1, we introduce the concept of strongly resid-ual h-integrability with exponent r concerning the array of constants {ani}as follows.

600 Wang X. J., Hu S. H., Volodin A. I.

Definition 3.5. Let {Xni, un 6 i 6 vn, n > 1} be an array of randomvariables and let {ani, un 6 i 6 vn, n > 1} be an array of constants. Letr > 0. The array {Xni, un 6 i 6 vn, n > 1} is said to be strongly residu-ally h-integrable (SR-h-integrable, for short) with exponent r concerning thearray {ani, un 6 i 6 vn, n > 1} if

supn>1

vn∑i=un

|ani|rE|Xni|r < ∞

and∞∑n=1

vn∑i=un

|ani|rE(|Xni| − h1/r(n)

)rI(|Xni|r > h(n)

)< ∞.

When r = 1, the preceding definition reduces to the concept of SR-h-inte-grability concerning the array {ani, un 6 i 6 vn, n > 1}, which was intro-duced by Ordonez Cabrera et al. [18].

The main idea of the notion of SR-h-integrability with exponent r con-cerning the array {ani, un 6 i 6 vn, n > 1} is to deal with weighted sums ofrandom variables. We introduce a new concept of integrability which dealswith usual normed sums of random variables as follows.

Definition 3.6. Let {Xni, un 6 i 6 vn, n > 1} be an array of randomvariables and r > 0. The array {Xni, un 6 i 6 vn, n > 1} is said to beSR-h-integrable with exponent r if

supn>1

1

kn

vn∑i=un

E|Xni|r < ∞

and∞∑n=1

1

kn

vn∑i=un

E(|Xni| − h1/r(n)

)rI(|Xni|r > h(n)

)< ∞.

Remark 3.6. It is easily seen that the concept of SR-h-integrability withexponent r is stronger than the concept of R-h-integrability with exponent r.

Our main result on strong convergence for weighted sums of arrays of row-wise m-NOD random variables under some uniformly integrable conditionsis as follows.

Theorem 3.3. Let 1 6 r < 2. Let {Xni, un 6 i 6 vn, n > 1} be an arrayof rowwise m-NOD random variables and let {ani, un 6 i 6 vn, n > 1} bean array of constants. Assume that the following conditions hold:

(i) {Xni, un 6 i 6 vn, n > 1} is SR-h-integrable with exponent r concern-ing the array {ani, un 6 i 6 vn, n > 1};

(ii)∞∑n=1

(h(n) sup

un6i6vn

|ani|r)(2−r)/r

< ∞.

Moment inequalities for m-NOD random variables and their applications 601

Then∑vn

i=unani(Xni −EXni) → 0 a.s. as n → ∞.

Proof. We use the same notation as those in Theorem 3.1. To prove∑vni=un

ani(Xni −EXni) → 0 a.s. as n → ∞, it suffices to show that

Sn.=

vn∑i=un

ani(Yni −EYni) → 0 a.s. as n → ∞ (3.1)

and

Tn.=

vn∑i=un

ani(Zni −EZni) → 0 a.s. as n → ∞. (3.2)

Firstly, we will prove (3.1). Note that |Yni| = min{|Xni|, h1/r(n)}, we haveby Markov’s inequality, Theorem 2.1 (or Remark 2.1), Jensen’s inequality,and conditions (i), (ii) that for any ε > 0,

∞∑n=1

P(|Sn| > ε) 61

ε2

∞∑n=1

E

∣∣∣∣ vn∑i=un

ani(Yni −EYni)

∣∣∣∣2 6 C∞∑n=1

vn∑i=un

a2niEY 2ni

6 C∞∑n=1

h(2−r)/r(n) supun6i6vn

|ani|2−rvn∑

i=un

|ani|rE|Yni|r

6 C

∞∑n=1

[h(n) sup

un6i6vn

|ani|r](2−r)/r

(supn>1

vn∑i=un

|ani|rE|Xni|r)

< ∞,

which implies (3.1) by Borel–Cantelli lemma.In the following, we will prove (3.2). Note that

|Zni| = (|Xni| − h1/r(n))I(|Xni|r > h(n)

),

we have by Markov’s inequality, Theorem 2.1 (or Remark 2.1), Jensen’s in-equality and condition (i) that for any ε > 0,

∞∑n=1

P(|Tn| > ε) 61

εr

∞∑n=1

E

∣∣∣∣ vn∑i=un

ani(Zni −EZni)

∣∣∣∣r 6C∞∑n=1

vn∑i=un

|ani|rE|Zni|r

= C∞∑n=1

vn∑i=un

|ani|rE(|Xni| − h1/r(n)

)rI(|Xni|r > h(n)

)< ∞,

which implies (3.2) by Borel–Cantelli lemma. This completes the proof ofthe theorem.

If we take ani = k−1/rn for un 6 i 6 vn and n > 1 in Theorem 3.3, then

we can get the following result on strong convergence for arrays of rowwisem-NOD SR-h-integrable with exponent 1 6 r < 2 random variables.

602 Wang X. J., Hu S. H., Volodin A. I.

Corollary 3.4. Let {Xni, un 6 i 6 vn, n > 1} be an array of row-wise m-NOD SR-h-integrable random variables with exponent 1 6 r < 2,kn → ∞, h(n) ↑ ∞, and

∞∑n=1

(h(n)

kn

)(2−r)/r

< ∞.

Then1

k1/rn

vn∑i=un

(Xni −EXni) → 0 a.s.

as n → ∞.

4. On the asymptotic approximation of inverse moment fornonnegative m-NOD random variables. As one application of themoment inequalities for m-NOD random variables, Section 3 deals with theLr convergence and strong convergence for m-NOD random variables undersome uniformly integrable conditions. As another application of the momentinequalities for m-NOD random variables, we will study the asymptotic ap-proximation of inverse moments for nonnegative m-NOD random variableswith finite first moments in this section.

Let Z1, Z2, . . . be a sequence of nonnegative random variables with finitesecond moments. Denote

Xn =1

Bn

n∑i=1

Zi, B2n =

n∑i=1

D(Zi). (4.1)

Under some suitable conditions, the inverse moment can be approximatedby the inverse of the moment in the following way:

E(a+Xn)−α ∼ (a+EXn)

−α, (4.2)

where a > 0 and α > 0 are arbitrary real numbers. Here and in what follows,for two positive sequences {cn, n > 1} and {dn, n > 1}, we write cn ∼ dnand cn = o(dn) if limn→∞ cnd

−1n = 1, limn→∞ cnd

−1n = 0. The left-hand side

of (4.2) is the inverse moment and the right-hand side is the inverse of themoment. Usually, the inverse of the moment is much easier to compute thanthe inverse moment. So in many practical applications, such as evaluatingrisks of estimators and powers of tests, reliability, life testing, insurance,and financial mathematics, complex systems, and so on, we often take theinverse of the moment instead of the inverse moment. Up to now, manyauthors studied the asymptotic approximation of inverse moment and foundmany interesting results. For the details about the inverse moment, one canrefer to [5], [7], [8], [15], [43], [37], [26], [24], [47] [9], [27], and etc.

Moment inequalities for m-NOD random variables and their applications 603

For n > 1, denote

Xn =n∑

i=1

Zi, µn = EXn (4.3)

and

µn,s =n∑

i=1

EZiI(Zi 6 µsn/

√n) for some 0 < s < 1.

Based on notation above, Shi et al. [26] obtained the following Theorem Cand Theorem D.

Theorem C. Let {Zn, n > 1} be a sequence of independent, nonnegative,and nondegenerated random variables. Assume that the following conditionshold:

(H1) µn → ∞ as n → ∞;(H2) µn ∼ µn,s for some 0 < s < 1.Then

E(a+ Xn)−α ∼ (a+EXn)

−α (4.4)

holds for all real constants a > 0 and α > 0.Theorem D. Let the conditions of Theorem C hold. In addition, suppose

that there exists a function f(x), x > 0, satisfying the following conditions:(H3) there exists a c1 > 0 such that f(x) > c1 for x > 0;(H4) there exist k > 0 and c2 > 0 such that f(x)/xk → c2 as x → ∞;(H5) 1/f(x) is a convex function for x > 0.Then

E[f(Xn)]−1 ∼ [f(EXn)]

−1. (4.5)

Denote

µn,s =

n∑i=1

EZiI(Zi 6 µsn) for some 0 < s < 1.

Consider the following assumption:(H6) µn ∼ µn,s for some 0 < s < 1.Yang et al. [47] pointed out that condition (H6) is weaker than (H2) and

extended Theorem C for independent random variables to the case of non-negative random variables under conditions (H1) and (H6).

Recently, Shen [24] generalized the result of Theorem C to a general caseand obtained the following result.

Theorem E. Let {Zn, n > 1} be a sequence of nonnegative random vari-ables with EZn < ∞ for all n > 1 and 0 < s < 1. Let {Mn, n > 1} and{an, n > 1} be sequences of positive constants such that an > C for all n suf-ficiently large, where C is a positive constant. Denote Xn = M−1

n

∑nk=1 Zk

and µn = EXn and Dn = ηMnµsn/an , where η is a positive constant. Sup-

pose that the following conditions hold:

604 Wang X. J., Hu S. H., Volodin A. I.

(i) For any p > 2, there exist positive constants η and C (depending onlyon p) such that

E

∣∣∣∣ n∑k=1

(Z ′nk −EZ ′

nk)

∣∣∣∣p 6 C

[ n∑k=1

E|Z ′nk −EZ ′

nk|p +( n∑

k=1

Var(Z ′nk)

)p/2],

where Z ′nk = ZkI(Zk 6 Dn) +DnI(Zk > Dn), or Z ′

nk = ZkI(Zk 6 Dn);(ii) µn → ∞ as n → ∞;(iii) ∑n

k=1EZkI(Zk > Dn)∑nk=1EZk

→ 0 as n → ∞,

where η > 0 is the same as that in (i).Then (4.2) holds for all real constants a > 0 and α > 0.Inspired by the literatures above, we will establish the asymptotic approx-

imation of inverse moment as follows.Theorem 4.1. Let the conditions of Theorem E and (H3)–(H5) hold. In

addition, assume that there exists a positive constant γ such that f(x) is anondecreasing function for x > γ . Then

E[f(Xn)]−1 ∼ [f(EXn)]

−1. (4.6)

Proof. Applying Jensen’s inequality to the convex function 1/f(x), wehave

E[f(Xn)]−1 > [f(EXn)]

−1,

which implies that

lim infn→∞

f(EXn)E[f(Xn)]−1 > 1. (4.7)

To prove (4.6), we only need to show

lim supn→∞

f(EXn)E[f(Xn)]−1 6 1. (4.8)

For any 0 < δ < 1, let

Un = M−1n

n∑k=1

[ZkI(Zk 6 Dn) + ηMnµ

sn/anI(Zk > Dn)

] .= M−1

n

n∑k=1

Z ′nk

and

E[f(Xn)]−1 = E[f(Xn)]

−1I(Un>µn − δµn) +E[f(Xn)]−1I(Un<µn − δµn)

.= Q1 +Q2. (4.9)

Moment inequalities for m-NOD random variables and their applications 605

Note that Xn > Un and f(x) is a nondecreasing function for x > γ, we haveby (H4) that

lim supn→∞

f(EXn)Q1 6 lim supn→∞

f(µn)E[f(Un)]−1I(Un > µn − δµn)

6 lim supn→∞

[f(µn)

µkn

· µkn

(µn − δµn)k· (µn − δµn)

k

f(µn − δµn)

]= (1− δ)−k → 1 as δ ↓ 0. (4.10)

In the following, we will prove that

limn→∞

f(EXn)Q2 = 0. (4.11)

For 0 < δ < 1 given above, it follows by (iii) in Theorem E that there existspositive integer n(δ) > 0 such that

n∑k=1

EZkI(Zk > Dn) 6δ

4

n∑k=1

EZk, n > n(δ), (4.12)

which implies that for n > n(δ),

|µn −EUn| =∣∣∣∣M−1

n

n∑k=1

EZkI(Zk > Dn)−M−1n

n∑k=1

DnEI(Zk > Dn)

∣∣∣∣6 M−1

n

n∑k=1

EZkI(Zk > Dn) +M−1n

n∑k=1

DnEI(Zk > Dn)

6 M−1n

n∑k=1

EZkI(Zk > Dn) +M−1n

n∑k=1

EZkI(Zk > Dn)

= 2M−1n

n∑k=1

EZkI(Zk > Dn) 6δµn

2. (4.13)

By condition (H3), (4.13), Markov’s inequality, condition (i) in Theorem E,and Cr-inequality, we have for any p > 2 and all n > n(δ) that

Q2 6 CP

(|Un −EUn| >

δµn

2

)6 Cµ−p

n M−pn E

∣∣∣∣ n∑k=1

(Z ′nk −EZ ′

nk)

∣∣∣∣p6 Cµ−p

n

(M−2

n

n∑k=1

EZ2kI(Zk 6 Dn) +M−2

n

n∑k=1

D2nEI(Zk > Dn)

)p/2

+ Cµ−pn

[M−p

n

n∑k=1

EZpkI(Zk 6 Dn) +M−p

n

n∑k=1

DpnEI(Zk > Dn)

]

606 Wang X. J., Hu S. H., Volodin A. I.

6 Cµ−pn

(M−1

n

µsn

an

n∑k=1

EZkI(Zk 6Dn)+M−1n

µsn

an

n∑k=1

EZkI(Zk >Dn)

)p/2

+ Cµ−pn M−1

n

µs(p−1)n

ap−1n

n∑k=1

EZkI(Zk 6 Dn)

+ Cµ−pn M−1

n

µs(p−1)n

ap−1n

n∑k=1

EZkI(Zk > Dn)

= C

[µ−(1−s)p/2n

ap/2n

+µ−(1−s)(p−1)n

ap−1n

]6 Cµ−(1−s)p/2

n +Cµ−(1−s)(p−1)n . (4.14)

Note that p > 2, and thus p − 1 > p/2. Taking p > max{2, 2k/(1 − s)},we have by (4.14) that Q2 = o(µ−k

n ), which together with (H4) yield (4.11).Hence, (4.8) follows by (4.9)–(4.11) immediately.

Following similar arguments, we can get (4.6) easily for

Un = M−1n

n∑k=1

ZkI(Zk 6 Dn).= M−1

n

n∑k=1

Z ′nk.

This completes the proof of the theorem.Remark 4.1. Since the Rosenthal type inequality (i.e., condition (i) in The-

orem E) is satisfied for m-NOD random variables, the result of Theorem 4.1holds for nonnegative m-NOD random variables and other random variables,such as ρ-mixing random variables, ϕ-mixing random variables, ρ-mixingrandom variables, NA random variables, NSD random variables, NOD ran-dom variables, END random variables, AANA random variables, ρ−-mixingrandom variables, and so on.

Remark 4.2. Take f(x) = (a + x)α, x > 0, a > 0, and α > 0. It is easyto check that f(x) is a nondecreasing function for x > 0, and conditions(H3)–(H5) hold. Hence, Theorem E can be obtained by Theorem 4.1 easily.That is to say, Theorem E is a special case of Theorem 4.1. In addition, if wetake Mn ≡ 1 and an = η

√n, then we can get Theorem D immediately; if we

take Mn ≡ 1 and an = η, then we can get Theorem 2.3 of [47] immediately; ifwe take an = usn, then we can get Theorem 2.2 of [9] immediately. Therefore,our result of Theorem 4.1 generalize the corresponding one of [26], [24], [47],and [9].

Acknowledgments. The authors are most grateful to the editor andanonymous referee for careful reading of the manuscript and valuable sug-gestions which helped in significantly improving an earlier version of thispaper.

Moment inequalities for m-NOD random variables and their applications 607

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Поступила в редакцию31.III.2015