Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 104390,10 pages doi:10.1155/2012/104390

Research Article

Sufficient and Necessary Conditions of Complete Convergence for Weighted Sums of PNQD Random Variables

Qunying Wu1,2

1 College of Science, Guilin University of Technology, Guilin 541004, China

2 Guangxi Key Laboratory of Spatial Information and Geomatics, Guilin University of Technology, Guilin 541004, China

Correspondence should be addressed to Qunying Wu, wqy666@glite.edu.cn

Received 7 February 2012; Accepted 19 April 2012

Academic Editor: Martin Weiser

Copyright © 2012 Qunying 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.

The complete convergence for pairwise negative quadrant dependent (PNQD) random variables is studied. So far there has not been the general moment inequality for PNQD sequence, and therefore the study of the limit theory for PNQD sequence is very difficult and challenging. We establish a collection that contains relationship to overcome the difficulties that there is no general moment inequality. Sufficient and necessary conditions of complete convergence for weighted sums of PNQD random variables are obtained. Our results generalize and improve those on complete convergence theorems previously obtained by Baum and Katz (1965) and Wu (2002).

1. Introduction and Lemmas

Random variables X and Y are said to be negative quadrant dependent (NQD) if

P(X < x,Y < y) < P(X < x)P(Y < y), (1.1)

for all x,y e R. A sequence of random variables {Xn; n > 1} is said to be pairwise negative quadrant dependent (PNQD) if every pair of random variables in the sequence is NQD. This definition was introduced by Lehmann [1]. Obviously, PNQD sequence includes many negatively associated sequences, and pairwise independent random sequence is the most special case.

In many mathematics and mechanic models, a PNQD assumption among the random variables in the models is more reasonable than an independence assumption. PNQD series have received more and more attention recently because of their wide applications

in mathematics and mechanic models, percolation theory, and reliability theory. Many statisticians have investigated PNQD series with interest and have established a series of useful results. For example, Matula [2], Li and Yang [3], and Wu and Jiang [4] obtained the strong law of large numbers, Wang et al. [5] obtained the Marcinkiewicz's weak law of large numbers, Wu [6] obtained the strong convergence properties of Jamison weighted sums, the three-series theorem, and complete convergence theorem, and Li and Wang [7] obtained the central limit theorem. It is interesting for us to extend the limit theorems to the case of PNQD series. However, so far there has not been the general moment inequality for PNQD sequence, and therefore the study of the limit theory for PNQD sequence is very difficult and challenging. In the above-mentioned conclusions, only the Kolmogorov-type strong law of large numbers obtained by Matula [2, Theorem 1] and Baum and Katz-type complete convergence theorem obtained by Wu [6, Theorem 4] achieve the corresponding conclusions of independent cases, and the rest did not achieve the optimal results of independent cases.

Complete convergence is one of the most important problems in probability theory. Recent results of the complete convergence can be found in Wu [6], Chen and Wang [8], and Li et al. [9]. In this paper, we establish a collection that contains relationship to overcome the difficulties that there is no the general moment inequality and obtain the complete convergence theorem for weighted sums of PNQD sequence, which extend and improve the corresponding results of Baum and Katz [10] and Wu [6].

Lemma 1.1 (see [1]). Let X and Y be NQD random variables. Then

(i) cov(X, Y) < 0,

(ii) P(X > x,Y > y) < P(X > x)P(Y > y), for all x,y e R,

(iii) if f and g are Borel functions, both of which being monotone increasing (or both are monotone decreasing), then f (X) and g(Y) are NQD.

Lemma 1.2 (see [6, Lemma 2]). Let {Xn; n > 1} be a sequence of PNQD random variables with EXn = 0, EXn < go, Tj(k)= jk+i Xi, j > 0. Then

E(Tj(k))2 <2 EX2, i=j+1

A 1 2 j+n

Emax (Tj (k))2 < ^ £eX2. 1<k<" log2 2 j i

Lemma 1.3 (see [2, Lemma 1]). (i) IfZP(An) < to, then P(An; i.o.) = 0.

(ii) if P(AkAm) < P(Ak)P(Am), k = m, and P(An) = to, then P(An; i.o.) = 1.

Lemma 1.4. Let {Xn; n > 1} be a sequence of PNQD random variables. Then for any x > 0, there exists a positive constant c such that for all n > 1,

1 - p(max|Xk| >x\) VP(|Xk| > x) < cp(max|Xk| > x\ (1.3)

\1<k<n // k=1 \1<k<n /

Proof. We can prove the Lemma by Lemma A.6 of Zhang and Wen [11]. □

Journal of Applied Mathematics 2. Main Results and the Proof

In the following, the symbol c stands for a generic positive constant which may differ from one place to another. Let an << bn (an » bn) denote that there exists a constant c > 0 such that an < cbn (an > cbn) for all sufficiently large n, and let Xi < X (Xi > X) denote that there exists a constant c > 0 such that P(|x| > x) < cP(|X| > x) (P(|x| > x) > cP(|X| > x)) for all i > 1 and x > 0.

Theorem 2.1. Let {Xn; n > 1} be a sequence ofPNQD random variables with Xi < X. Let {ank; k < n,n > 1} be a sequence of real numbers such that

ank | < n a, vk < n, n > 1.

Let for ap> 1, 0 <p < 2, a> 0, and EXi = 0,for a < 1.If

exf < go,

fnap-2P(max|Snk| >e}< g, Vs> 0, (2.3)

\1<k<n /

n=1 x < < '

where Snk ^ E¿=i anX

Theorem 2.2. Let {Xn; n > 1} be a sequence ofPNQD random variables with Xi > X. Let {ank; k < n, n > 1} be a sequence of real numbers such that |ank| » n-a, for all k < n, n > 1. Let for a > 0, ap> 1, 0 <p< 2. If (2.3) holds, then (2.2) holds.

Remark 2.3. Taking ani = n-a, for all i < n, n > 1 in Theorem 2.1, then

G / \ G /

Yjiav-1P\max|Snk| >e) ^nap-2p( maxn

n=1 \1<k<n / n=1 \

Ynav-1 p(

i=1 Xi

Hence, Theorem 4 in Wu [6] is a particular case of our Theorem 2.1.

Remark 2.4. When {Xn; n > 1} is i.i.d. and ani = n-a, for all i < n, n > 1, then Theorems 2.1 and 2.2 become Baum and Katz [10] complete convergence theorem. Hence, our Theorems 2.1 and 2.2 improve and extend the well-known Baum and Katz theorem.

Proof of Theorem 2.1. Without loss of generality, assume that ank > 0for k < n, n > 1. Let q> 0 such that (1 + (1/ap))/2 <q< 1. For all i < n, let

Ym = - a-1 na(q-1)^amXl < -na(q-1)) + Xtl(am|Xi| < na(q-1))

+ a-1 na(q-V)l(amXi>na(q-V)),

Unk ^ ^aniYni.

An = U (WnjXj I > e), /=1

j (2.6)

Bn = U ((amXr > na(q-1),anjXj > na(q-1)){J[aniXl < -na(q-1),an/Xj < -na(q-1))).

1<i<j<n

Firstly, we prove that

(max|Snk| < 6e) D Ac„ Ofmax|Unk| < 2e) O Bcn

1<k<n n 1<k<n n

= Q(IanjXjI < e) f| (^axJUnk| < 2e) f) 0 [{{anX < na(q-1))U(an/Xj < na(q-1)))

1<i<j<n

0 ((anX > -na(q-1)) H(a„jXj > -na(q-1)))]

For any w e Dn, we have

IanjXjI < e, IanjYn/1 < IanjX/1 < e, V1 < j < n, max|Unk| < 2e, (2.8)

1<k<n ^ ' '

and for any 1 < i < j < n,

aniXi < na(q-1), or anjXj < na(q-1\ aniXi >-na(q-1), or a„jXj >-na(q-1).

Journal of Applied Mathematics Hence

a=#{i; 1 < i < n,aniXi(w) > na(q-1)} < 1, &=#{i; 1 < i < n, aniXi(w) < -na(q-1)} < 1,

(2.10)

where the symbol denotes the number of elements in the set A.

When a = b = 0, then a^X^M) < na(q-1) for any 1 < i < n; thus, Y^(m) = Xi(M), and therefore by (2.8),

max|Snk| = max|Unk| < 2e < 6e.

1< k<n 1< k<n

(2.11)

When a = 1, b = 0 (or a = 0, b = 1), then there exists only an i0: 1 < i( < n such that an(Xi0 (m) > na(q-1) (or ani0X^ (m) < -na(q-1)), the remaining j, ajXnj (m)| < na(q-1); thus, Xj(m) = Ynj(m). If 1 < k < i0 - 1, then Snk(m) = Unk(m). If i( < k < n, then by (2.8),

max |Snk (w)| = max

1<k<n 1<k<n

^ aniXi(w) + anio Xio (w) 1<i<k,i / i0

1< k<n

^aniYni(w) an0 Yni0 (w) + ani0 Xi0 (w)

(2.12)

1< k<n

J^aniYni(w)

< 2e + e + e < 6e.

+ |anio Ynio (w)| + |anio Xio (w)|

When a = b = 1, then there exist 1 < i1, i2 < n such that ani1 Xi1 (m) > na(q-1), ani2Xi2(m) < -na(q-1), the remaining j, a^Xj(m)| < na(q-1); thus, Xj(m) = Ynj(m). Without loss of generality, assume that i1 < i2. If 1 < k < i1 - 1, then Snk(m) = Unk(m); if i1 < k< i2, then by (2.8),

max | Snk (w)| < max | Unk (w)| + |ani1 Yni1 (w)| + |ani1 Xi1 (w)|

< 2e + e + e < 6e.

If k > i2, then by (2.8),

(2.13)

max|Snk (w)| = max

1<k<n 1<k<n

^ aniXi(w) + ani1 Xi1 (w) + an2 Xh (w)

1<i<k,i / hi

< max|Unk(w)| + |ani1 Yni1 (w)| + |ani2Yni2(w)|

+ |ani1 Xi1 (w)| + |ani2 Xi2 (w)| < 6e.

(2.14)

Hence, (2.7) holds, that is:

(max |Snk | > 6e ) c An M ( max|Unk | > 2e ) M Bn. (2.15)

1<k<n 1<k<n

Therefore, in order to prove (2.3), we only need to prove that

£nttp-2P(An) < g, (2.16)

Yrn^P(Bn) < g, (2.17)

£nap-2p{max|Unk| > 2e) < g, ve > 0. (2.18)

n=1 1<k<n

By (2.1), (2.2), Xi < X, and ap > 1,

Yrnap-2P(An) <£nap-2£P(IanjXjI > e)

n=1 n=1 j=1

ea-1 ecn

g n / \

<Xnap-2XP(IXjI > ea-1 > ecna)

n=1 j=1

« ^nap1 P(|X| > ecna)

= X nap-^ P(ecja < |X| < ecj + 1)a) (2.19)

n=1 j=n

= XZnap-1P(ecja <|X| <ecj + 1)a)

j=1 n=1

< X japP(ecja < |X| < ecj + 1)a) j=1

« E|X|p < g.

That is, (2.16) holds.

By Lemma 1.1 (ii), Xi < X, and the definition of q, ap(1 - 2q) < -1,

|>ap-2P(Bn) < £nap-2 ^ (p(^aniXi > na(q-1))^an/Xj > na(q-1))

n=1 1<i<j<n

+ P^aniXi < -na(q-1))P^anjXj < -na(q-1)))

<< ^napP2(|X| > cnaq) < ^napn-2apq(E|X|p)2

<< ^nap(1-2q) <

(2.20)

That is, (2.17) holds.

In order to prove (2.18), firstly, we prove that

max EUnk | = max

1<k<n 1<k<n

E ani Yn

■ 0, n -> G.

(2.21)

(i) When a < 1, then p > 1/a > 1; from EXi = 0 and the definition of q, we have q < 1, apq > ap + 1 - apq = 1 + ap(1 - q) > 1 :

1< k<n

E ani Yn

<£ara|EYra| = £an^E(X, - Yra)|

< £ am{E\Xi + an1na(q-1) |l(a„iXi<-na(q-1)) + E|Xi - a-1na(q-1) |l( i=1

n n / a^X^ \ f-1

<<X^i^E-Xi^QaniX^n^q-1')) aniE|XiK ni i

L(aniXi>na(q-1))

na(q-1)

= X aniE|Xi|pna(1-q)(f-1)

^ n~ap+1+ap-a-apq+aq = n~(apq-1)-a(1-q)

-> 0, n -> 00.

(2.22)

(ii) When a> 1, and p > 1, then E|X| < g from (2.2), thus,

< ^aniEX < n-a+1 0, n

E ani Yn i=1

(2.23)

Journal of Applied Mathematics (iii) When a> 1, and p < 1,by-(ap-1) -a(1 -q)(1 -p) < 0, and -a(1 -q) - (apq-1) < 0,

we get

^an{Yn

< £an^E|Xi|I(a„i|Xi|<na(q-1)) + a-1 na(q-1)p(|araX^ > na(q-1)))

n n (2 24)

<2alEXi^|amXi|1-pI(an|Xr<na(q-l)^)+Zn^anrn-^EXf ( . ) i=1 i=1

« n-(ap-1)-a(1-p)(1-q) + n-a(1-q)-(«pq-D _> 0.

Hence, (2.21) holds; that is, for any e> 0, we have max^^EU^| < e for all sufficiently large n. Thus,

P ^ m ax I Unj I > 2e^ < ^rncixIUnj - EUnjI > e^. (2.25)

Let Yni = Yni - EYni. Obviously, Yni is monotonic on Xi. By Lemma 1.1 (iii), {Yni; n > 1,i < n} is also a sequence of PNQD random variables with EYni = 0, by Lemma 1.2 and -1 - a(1 - q)(2 - p) < -1:

Ynap-2P[ maxIUni - EUnA > .

n=1 Wn1 j jl

n=1 j=1

«£ nap-2log2^Ea2„jY2nj

« £nap-2log2^{Ea2yi)X1)Iajnl^q-ry)+ n2a(q-1)P(an,|X,| > na(q-1)))

n=1 j=1

(2.26)

< £ nap-2log2nX (E^njXj |pna(q-1)(2-p) + n2a(q-1)-ap(q-1)E| anjXj|p)

n=1 j=1

« X (nap-1-ap+a(q-1)(2-p) + n-1+ap-apq+2aq-2a^j log2n

= ^ n-1-a(1-q)(2-p)log2n

n=1 <00.

This completes the proof of Theorem 2.1. □

Proof of Theorem 2.2. Noting that max^^ ankXk | < 2max1<k<n|Snk | and |ank | » n-a, from (2.3),

£nap-2p(max|Xk| > ma\ < g, V£> 0. (2.27)

n=1 1<k<n

Thus, by ap - 2 > -1, we get

Vpf max |Xk| > £2a(j+1A < Y.n^pf max|Xk| > £na) < g, V£ > 0. (2 28)

\1<k<2J / n=1 \1<k<n / v ' y

This implies that

max p(max\X; \ > £22an^ < p( max X \ > £2a(m+1A 0. (2.29)

2m-1<n<2m \Kj<n' / \1<j<2ml /

Hence, for all sufficiently large n,

P^inax \ Xj \ > £22an^ < 2. (2.30)

By Lemma 1.4,

VP(|Xk| > £na) < 4cP(^max|Xk| > £na\ V£ > 0, (2.31)

k= \1<k<n /

which together with (2.27),

£nap-2£P(|Xk| >£na) < g, V£> 0. (2.32)

n=1 k=1

By Xk > X, we obtain

E|X|p << £nap-1P(|X| > £na) < G. (2.33)

This completes the proof of Theorem 2.2. □

Acknowledgments

The author is very grateful to the referees and the editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. Supported by the National Natural Science Foundation of China (11061012), and 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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