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Strong convergence of pairwise NQD random sequences Rui Li a,∗ , Weiguo Yang b,1 a Department of Mathematics, Taiyuan Normal College, Taiyuan 030012, China b Faculty of Science, Jiangsu University, Zhenjiang 212013, China

Received 24 December 2007 Available online 7 March 2008 Submitted by V. Pozdnyakov

Abstract Strong limit theory is one of the most important problems in probability theory. Some results on the convergence of pairwise NQD random sequences have been presented. This paper further analyzes the strong convergence of pairwise NQD sequences and generalizes partial results of Wu [Q.Y. Wu, Convergence properties of pairwise NQD random sequences, Acta Math. Sinica 45 (3) (2002) 617–624 (in Chinese)]. Since no general moment inequalities are given as so far, we avoid this problem and obtain a class of strong limit theorem for NQD sequences and some corresponding conclusions by use of truncation methods and generalized three series theorem, which are the supplements to the previous fruits. © 2008 Elsevier Inc. All rights reserved. Keywords: Pairwise NQD random variables; Strong convergence; Generalized three series theorem; Cr-inequality

1. Induction Definition 1. The pair (X, Y ) of random variables X and Y is said to be NQD (negatively quadrant dependent), if ∀x, y ∈ R, P (X x, Y y) P (X x)P (Y y). A sequence of random variables {Xn , n 1} is said to be pairwise NQD, if {Xi , Xj } is NQD for every i = j , i, j = 1, 2 . . . . This definition given by Lehmann [3] in 1966 has been applied widely. Obviously, pairwise NQD sequence includes many negatively associated sequences, and pairwise independent random sequence is the most special case. Since strong limit theory is one of the most classical problems in probability theory, many scholars have dedicated themselves to the study of it. For example, Jardas [2] and Yang [8] studied the strong limit theorems of arbitrary random variables, Matula [4] gained the Kolmogorov-type strong law of large numbers for the identically distributed pairwise NQD sequences, Chen [1] discussed Kolmogorov–Chung strong law of large numbers for the non-identically distributed pairwise NQD sequences under very mild conditions, Wu [5] gave the three series theorem of pairwise NQD sequences and proved the Marcinkiewicz strong law of large numbers. The authors take the inspiration in [2,7] * Corresponding author.

E-mail address: [email protected] (R. Li). 1 Supported by the National Science Foundation of China (10571076).

0022-247X/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2008.02.053

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R. Li, W. Yang / J. Math. Anal. Appl. 344 (2008) 741–747

and discuss the strong convergence of pairwise NQD sequences by applying truncation methods and generalized three series theorem, which extends partial results of Wu [5]. In this paper, we denote Xna −aI(Xn <−a) + Xn I(|Xn |a) + aI(Xn >a) , a > 0, and “” as general “O.” Lemma 1. (See [7].) Let {sn , n 1} and {tn , n 1} be non-negative number sequences, sn tn for each n 1, then the infinite series ∞

|un |sn < +∞

⇒

n=1

∞

|un |tn < +∞.

n=1

Lemma 2. (See [6].) Let X and Y be the NQD random variables, then (1) E(XY ) EXEY . (2) P (X > x, Y > y) P (X > x)P (Y > y). (3) If (r(·), s(·)) is a non-decreasing (or non-increasing) function, then r(X) and s(Y ) are still NQD random variables. Lemma 3 (Generalized three series theorem). (See [6].) Let {Xn , n 1} be pairwise NQD sequences. For some c > 0, we denote Xnc −cI(Xn <−c) + Xn I(|Xn |c) + cI(Xn >c) . If ∞ P |Xn | > c < ∞,

∞

n=1

n=1

EXnc < ∞,

∞

log2 n Var Xnc < ∞,

n=1

then ∞

Xn converges a.s.

n=1

2. Mainstream Theorem 1. Let {Xn , n 1} be a sequence of pairwise NQD random variables (r.v.s) and EXn = 0. Let {an , n 1} be a positive number sequence such that an ↑ ∞. Suppose that ϕn : R+ → R+ be Borel functions and αn 1, βn 2, Kn 1, Mn 1 (n ∈ N ) be constants satisfying 0 < x1 x2

⇒

ϕn (x2 ) ϕn (x1 ) αn Kn x1 x2αn

and β

β

x1 n x n Mn 2 . ϕn (x1 ) ϕn (x2 )

(1)

If ∞

Kn log2 n

n=1 ∞

Eϕn (|Xn |) < ∞, ϕn (an )

(2)

Eϕn (|Xn |) < ∞, ϕn (an )

(3)

Mn log2 n

n=1

then n 1 Xk = 0 a.s. n→∞ an

lim

k=1

(4)

R. Li, W. Yang / J. Math. Anal. Appl. 344 (2008) 741–747

Proof. By applying Kronecker lemma, to prove (4), it suffices to show that From Lemma 2, simultaneously

Xn an

∞

743

Xn n=1 an

converges a.s.

is obviously still a sequence of pairwise NQD r.v.s. If the following three inequalities hold

∞ P |Xn | > an < ∞, n=1 ∞ n=1 ∞ n=1

(5)

EXnan < ∞, an

(6)

log2 n Var Xnan < ∞, an2

(7)

then, by Lemma 3 (let c = 1), we obtain (4). According to (1), α ϕn (x2 ) x2 x2 n , αn Kn x1 x1 ϕn (x1 )

and ∞ ∞ P |Xn | an = n=1

0 < x1 x2 , dP

(8)

∞

∞

|Xn | dP an

n=1 |X |a n=1 |X |a n=1 |X |a n n n n n n ∞ ∞ Eϕn (|Xn |) Eϕn (|Xn |) Kn Kn log2 n , ϕn (an ) ϕn (an ) n=1 n=1

Kn

ϕn (|Xn |) dP ϕn (an ) (9)

which together with (2) gives (5). Next, we prove (6). Since EXn = 0, and from (8), ∞ ∞ EX an = E(−an I(X n

n=1

n <−an )

+ Xn I(|Xn |an ) + aI(Xn >an ) )

n=1

=

∞ n=1

n=1 ∞

n=1 ∞

∞ n=1

∞

∞

n=1

Ean I(|Xn |>an ) +

∞

Ean I(|Xn |>an ) +

|EXn I(|Xn |an ) | |EXn I(|Xn |>an ) |

n=1 ∞

ϕn (|Xn |) I(|Xn |>an ) + Ean Kn ϕn (an )

∞

Eϕn (|Xn |) an Kn + ϕn (an ) an Kn

n=1 ∞

=2

n=1

Eϕn (|Xn |) + ϕn (an )

an Kn

n=1 |X |>a n n ∞

an Kn

n=1

|Xn | dP

n=1 |X |>a n n

ϕn (|Xn |) dP ϕn (an )

Eϕn (|Xn |) ϕn (an )

∞

Eϕn (|Xn |) Eϕn (|Xn |) 2 , an Kn an Kn log2 n ϕn (an ) ϕn (an ) n=1

in conjunction with (2), we have ∞ ∞ Eϕn (|Xn |) EX an 2 < ∞. an Kn log2 n n ϕn (an ) n=1

n=1

(10)

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R. Li, W. Yang / J. Math. Anal. Appl. 344 (2008) 741–747

Now, we prove (7). Noting that ∞ log2 n Var Xnan

an2

n=1

=

∞ log2 nE(Xnan − EXnan )2

an2

n=1

∞ log2 nE(Xnan )2

an2

n=1

=

∞ log2 nE(Xnan )2 n=1

an2

−

∞ log2 n(EXnan )2 n=1

an2 (11)

,

log2 nE(Xnan )2 we only prove the convergence of ∞ . n=1 an2 For any n ∈ N , by applying Cr-inequality, 2 E Xnan = E|−an I(Xn <−an ) + Xn I(|Xn |an ) + an I(Xn >an ) |2 3E an2 I(Xn <−an ) + Xn2 I(|Xn |an ) + an2 I(Xn >an ) E an2 I(|Xn |>an ) + Xn2 I(|Xn |an )

(12)

by (8), Ean2 I(|Xn |>an ) Ean2

|Xn | ϕn (|Xn |) Eϕn (|Xn |) I(|Xn |>an ) an2 Kn , I(|Xn |>an ) Ean2 Kn an ϕn (an ) ϕn (an )

furthermore, by use of (2), we get ∞

∞

log2 n

n=1

Ean2 I(|Xn |>an ) Eϕn (|Xn |) < ∞. Kn log2 n ϕn (an ) an2

(13)

n=1

On the other hand, again from (1), we easily know x12 x22

β

x1 n β x2 n

Mn

ϕn (x1 ) , ϕn (x2 )

x1 x2 ,

therefore, EXn2 I(|Xn |an ) =

Xn2 dP

|Xn |an

an2 Mn

|Xn |an

ϕn (|Xn |) Eϕn (|Xn |) dP an2 Mn , ϕn (an ) ϕn (an )

observing (3), ∞

∞

log2 n

n=1

EXn2 I(|Xn |an ) Eϕn (|Xn |) < ∞. Mn log2 n 2 ϕn (an ) an

(14)

n=1

By applying (12)–(14), ∞ log2 nE(Xnan )2 n=1

an2

< ∞,

(15)

which together with (11) gives (7). From what have been proved above, we obtain n 1 Xk = 0 a.s. n→∞ an

lim

k=1

2

R. Li, W. Yang / J. Math. Anal. Appl. 344 (2008) 741–747

745

Corollary 1. (See [5].) Let {Xn , n 1} be a sequence of pairwise NQD r.v.s, and {ϕn (x)} be a sequence of positive even functions which is defined on (0, ∞) satisfies: ϕnx(x) and ϕnx(x) are non-increasing, EXn = 0 (n ∈ N ). Moreover, 2 if 0 < an ↑ ∞ and ∞

log2 n

n=1

Eϕn (Xn ) < ∞, ϕn (an )

(16)

then n 1 Xk → 0, an

n→∞

a.s.

k=1

Proof. Taking αn = 1, βn = 2, Kn = Mn = 1 in Theorem 1, by applying of (1)–(3), we obtain x1 x2

⇒

ϕn (x1 ) ϕn (x2 ) , x1 x2

x12 x22 ϕn (x1 ) ϕn (x2 ) which gives ∞

∞

log2 n

n=1

Eϕn (Xn ) 2 Eϕn (|Xn |) = < ∞. log n ϕn (an ) ϕn (an ) n=1

Thus, the corollary follows.

2

Theorem 2. Let {Xn , n 1} be a sequence of pairwise NQD r.v.s and an be defined as in Theorem 1. Let ϕn : R+ → R+ be non-decreasing Borel functions satisfying p

x1 x2

⇒

p

x n x1 n 2 , ϕn (x1 ) ϕn (x2 )

(17)

when pn > 2 (n ∈ N), and if

∞ Eϕn (|Xn |) 1/pn log2 n < ∞, ϕn (an )

(18)

n=1

then n 1 Xk = 0 a.s. n→∞ an

lim

k=1

Proof. Similarly to Theorem 1, it suffices to show that (5)–(7) hold simultaneously. By Lemma 1 and (18), ∞ Eϕn (|Xn |) n=1

ϕn (an )

< ∞.

(19)

Analogously, ∞ ∞ P |Xn | an n=1

hence, (5) is verified.

n=1 |X |a n n

∞

Eϕn (|Xn |) ϕn (|Xn |) dP < ∞, ϕn (an ) ϕn (an ) n=1

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R. Li, W. Yang / J. Math. Anal. Appl. 344 (2008) 741–747

From (17) and Jensen inequality, we obtain ∞ ∞ EX an Ean I(|X n

n=1

n=1 ∞

n=1

∞

|EXn I(|Xn |an ) |

(20)

n=1 ∞

Ean

n=1 ∞

n |>an ) +

ϕn (|Xn |) I(|Xn |>an ) + E|Xn I(|Xn |an ) | ϕn (an ) n=1

∞

1/pn Eϕn (|Xn |) + an E |Xn I(|Xn |an ) |pn ϕn (an ) n=1

∞

∞

1/pn Eϕn (|Xn |) + an E|Xn I(|Xn |an ) |pn ϕn (an ) n=1 n=1 1/pn ∞ ∞

Eϕn (|Xn |) p Eϕn (|Xn |) an n + an ϕn (an ) ϕn (an ) n=1 n=1

∞ ∞ Eϕn (|Xn |) 1/pn Eϕn (|Xn |) + = an an ϕn (an ) ϕn (an ) n=1

(21)

n=1

obviously, owing to (18)–(20), ∞ |EXnan |

an

n=1

∞ Eϕn (|Xn |) n=1

ϕn (an )

+

∞

Eϕn (|Xn |) 1/pn n=1

ϕn (an )

< ∞.

So, (6) holds. From the assumption, we know Eϕn (|Xn |) EI(|Xn |>an ) = P |Xn | > an , ϕn (an ) then ∞

∞

log2 n

n=1

Ean2 I(|Xn |>an ) Eϕn (|Xn |) < ∞. log2 n 2 ϕn (an ) an

(22)

n=1

Moreover,

1/pn

1/pn EXn2 I(|Xn |an ) an E|Xn I(|Xn |an ) | = an E |Xn I(|Xn |an ) |pn an E|Xn I(|Xn |an ) |pn

1/pn

Eϕn (|Xn |) 1/pn p Eϕn (|Xn |) I(|Xn |an ) an an n an2 ϕn (an ) ϕn (an ) therefore, ∞ n=1

∞

log2 n

EXn2 I(|Xn |an ) Eϕn (|Xn |) < ∞. log2 n 2 ϕn (an ) an

(23)

n=1

Then, (12), (21) and (22) imply that ∞ log2 nE(Xnan )2 n=1

an2

< ∞.

By (11) and (13), (7) is proved. Thus, we complete the proof of this theorem.

(24)

2

Acknowledgments The authors express their gratitude to the referee for useful comments that helped to improve the initial version of this paper.

R. Li, W. Yang / J. Math. Anal. Appl. 344 (2008) 741–747

747

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