On the strong law of large numbers for weighted sums of arrays of rowwise negatively dependent random variables

J. Korean Math. Soc. 46 (2009), No. 4, pp. 827–840 DOI 10.4134/JKMS.2009.46.4.827

ON THE STRONG LAW OF LARGE NUMBERS FOR WEIGHTED SUMS OF ARRAYS OF ROWWISE NEGATIVELY DEPENDENT RANDOM VARIABLES

Jong-Il Baek, Hye-Young Seo, Gil-Hwan Lee, and Jeong-Yeol Choi

Abstract. Let {X

| 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise nega- tively dependent (N D) random variables. We in this paper discuss the conditions of P

i=1

a

X

→ 0 completely as n → ∞ under not neces- sarily identically distributed setting and the strong law of large numbers for weighted sums of arrays of rowwise negatively dependent random vari- ables is also considered.

1. Introduction

Let {X _n | n ≥ 1} be a sequence of random variables. Hsu and Robbins [4]

introduced the concept of complete convergence of {X n | n ≥ 1}. A sequence {X n | n ≥ 1} of random variables converges to a constant c completely if

X ∞ n=1

P (|X n − c| > ²) < ∞ for all ² > 0.

If X n → c completely, then the Borel-Cantelli lemma implies that X n → c almost surely, but the converse is not true in general.

Let {X ni | 1 ≤ i ≤ n, n ≥ 1} be an array of random variables with EX ni = 0 for all n and i. Many authors studied the complete convergence of n ^−1/p P _n

i=1 X ni which is defined (1.1)

X ∞ n=1

P Ã

|n ^−1/p X n i=1

X ni | > ²

!

for all ² > 0, where 0 < p < 2.

In particular, Erd¨os [4] showed that for an array of independent identically distributed (i.i.d.) random variables {X ni |1 ≤ i ≤ n, n ≥ 1}, (1.1) holds if and only if E|X 11 | ^2p < ∞. Hu et al. [6] showed that Erd¨os’ result could be obtained by replacing the i.i.d. condition by the uniformly bounded condition.

Received October 26, 2007.

2000 Mathematics Subject Classification. Primary 60F05; Secondary 62E10, 45E10.

Key words and phrases. complete convergence, Negatively dependent random variables, arrays, uniformly bounded random variable, strong convergence, weak convergence.

c

°2009 The Korean Mathematical Society

827

We recall that any away {X ni |1 ≤ i ≤ n, n ≥ 1} of random variables is said to be uniformly bounded by a random variable X if for all i, n and x ≥ 0, (1.2) sup P (|X ni | ≥ x) ≤ P (|X| > x).

Hu et al. [5] had obtained the following result in complete convergence and they had established (1.3) for non identcally random variable when no assump- tion of independence between rows of the array is made.

Theorem A. Let {X ni | 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise independent random variables with EX ni = 0. Suppose that {X ni | 1 ≤ i ≤ n, n ≥ 1}

are uniformly bounded by some random variable X. If E|X| ^2p < ∞ for some 1 ≤ p ≤ 2, then

(1.3) n ^−1/p

X n i=1

X _ni −→ 0 completely as n → ∞

if and only if E|X 11 | ^2p < ∞.

In this paper, we discuss the strong law of large numbers for weighted sums of arrays of rowwise N D random variables. The main purpose of this paper is to extend and generalize Theorem A to rowwise N D random variables which satisfy suitable conditions. Further, the last two properties of this paper show that neither N D nor EX ni = 0 are needed to obtain the corresponding strong law of large numbers when 0 < p < 1/2 or a weak law of large numbers when 1/2 ≤ p < 1.

2. Preliminaries

This section will list some background materials which will be used in ob- taining the main results in the next section and we define a ⁺ = max(0, a), a ⁻ = max(0, −a).

Definition 2.1 (Ebrahimi et al. [2]). Random variables X and Y are nega- tively dependent(N D) if

(2.1) P [X ≤ x, Y ≤ y] ≤ P [X ≤ x]P [Y ≤ y]

for all x, y ∈ R. A collection of random variables is said to be pairwise N D if every pair of random variables in the collection satisfies (2.1).

It is important to note that Definition 2.1 implies (2.2) P [X > x, Y > y] ≤ P [X > x]P [Y > y]

for the x, y ∈ R. Moreover, it follows that (2.2) implies (2.1), and hence, they are equivalent for pairwise N D. Ebrahimi and Ghosh [2] showed that (2.1) and (2.2) are not equivalent for n ≥ 3. Consequently, the following definition is needed to define sequences of negatively dependent random variables.

Definition 2.2. The random variables X 1 , X 2 , . . . are said to be

(a) lower negatively dependent(LN D) if for each n, (2.3) P [X 1 ≤ x 1 , . . . , X n ≤ x n ] ≤

Y n i=1

P [X i ≤ x i ] for all x 1 , . . . , x n ∈ R,

(b) upper negatively dependent(U N D) if for each n, (2.4) P [X 1 > x 1 , . . . , X n > x n ] ≤

Y n i=1

P [X i > x i ] for all x 1 , . . . , x n ∈ R,

(c) negatively dependent (N D) if both (2.3) and (2.4) hold.

The following properties are listed for reference in obtaining the main result in the next section and detailed proofs can be found in their paper.

Lemma 2.1 (Ebrahimi et al. [2]). Let {X n | n ≥ 1} be a sequence of N D random variables and {f n | n ≥ 1} be a sequence of monotone increasing (decreasing) Borel functions. Then {f n (X n ) | n ≥ 1} is a sequence of N D random variables.

Lemma 2.2 (Taylor et al. [7]). (a) Let X 1 , X 2 , . . . , X n be nonnegative random variables which are upper negatively dependent. Then

E Ã _n

Y

i=1

X i

!

≤ Y n i=1

EX i .

(b) Let X 1 , X 2 , . . . , X n be a pairwise N D random variables. Then (i) EX i X j ≤ EX i EX j for i 6= j

(ii) Cov(X i , X j ) ≤ 0 for i 6= j.

(c) Let X 1 , X 2 , . . . , X n be a N D random variables. Then for any real num- bers {a 1 , . . . , a n } and {b 1 , . . . , b n } such that a i ≤ b i , 1 ≤ i ≤ n, Y i = X i I (a i ≤ X _i ≤ b _i ) + b _i I(X _i > b _i ) + a _i I(X _i < a _i ) are N D random variables.

Lemma 2.3 (Hu et al. [6]). For any r ≥ 1 and p > 0, (a) E|X| ^r if and only if

X ∞ n=1

n ^r−1 P (|X| > ²n) < ∞ for all ² > 0,

(b) E|X| ^r I(|X| ≤ n ^1/p ) ≤ r Z _n

0

t ^r−1 P (|X| > t)dt,

(c) r2 ^−r X ∞ n=1

n ^r−1 P (|X| > n) ≤ E|X| ^r ≤ 1 + r2 ^r X ∞ n=1

n ^r−1 P (|X| > n).

Lemma 2.4 (Bozorgnia et al. [1]). Let X be a random variable such that EX = 0 and |X| ≤ M < a.e. Then for all constant M,

1 ≤ Ee ^tx ≤ e ^t

^EX

for all |t| ≤ 1 M .

3. Strong law of large numbers for weighted sums

Theorem 3.1 is to extend and generalize Theorem A to rowwise N D random variables. Note that the range 0 < p < 2 is allowed in Theorem 3.1 and Theorem 3.2 whereas previous results usually addressed the important subset 1 ≤ p ≤ 2. Throughout the proof c will represent positive constants whose value may change from one to another.

Theorem 3.1. Suppose that 0 < p < 2 and let {X ni | 1 ≤ i ≤ n, n ≥ 1}

be an array of rowwise N D random variables with EX ni = 0. Suppose that sup P (|X ni | > x) ≤ P (|X| > x) for all i, n and x ≥ 0. Assume that {a ni | 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers satisfying

(i) max

1≤x≤n |a ni | = O(n ^−1/p ), (ii) X n i=1

a ² _ni = o µ 1

log n

¶ .

If E|X| ^2p < ∞, then X n i=1

a ni X ni −→ 0 completely as n → ∞.

Proof. Since a ni = a ⁺ _ni − a ⁻ _ai , it suffices to show that (1)

X ∞ n=1

P ï ¯

¯ ¯

¯ X n i=1

a ⁺ _ni X ni

¯ ¯

¯ ¯

¯ > ²

!

< ∞ for all ² > 0,

(2)

X ∞ n=1

P ï ¯

¯ ¯

¯ X n i=1

a ⁻ _ni X ni

¯ ¯

¯ ¯

¯ > ²

!

< ∞ for all ² > 0.

We prove only (1), the proof of (2) is analogous.

To prove (1), we need only prove that (3)

X ∞ n=1

P Ã _n

X

i=1

a ⁺ _ni X ni > ²

!

< ∞ for all ² > 0,

(4)

X ∞ n=1

P Ã _n

X

i=1

a ⁺ _ni X ni < −²

!

< ∞ for all ² > 0.

Without loss of generality, we can assume that 0 < a ⁺ _ni ≤ n ^−1/p for all 1 ≤

i ≤ n, n ≥ 1 and let q be the constant such that 0 < p < q < 2p < 2 and for

α > 0, α = 1/p − 1/q.

Note that à _n

X

i=1

a ⁺ _ni X ni ≥ ²

!

⊂ Ã _n

X

i=1

a ⁺ _ni X ni I(|X ni | ≤ n ^1/q ) ≥ ²/2

!

[ (a ⁺ _ni X ni ≥ ²/2 for some i, 1 ≤ i ≤ n)

[ (X ni > n ^1/q for at least two values of i, 1 ≤ i ≤ n).

Thus,

(5)

X ∞ n=1

P Ã _n

X

i=1

a ni + X ni ≥ ²

!

≤ X ∞ n=1

P Ã _n

X

i=1

a ni + X ni I(|X ni | ≤ n ^1/q ) ≥ ²/2

!

+ X ∞ n=1

P (a ni + X ni ≥ ²/2 for some i, 1 ≤ i ≤ n)

+ X ∞ n=1

P (X ni > n ^1/q for at least two values of i, 1 ≤ i ≤ n)

= I 1 + I 2 + I 3 (say).

To prove I 1 < ∞, we first define that

Y _ni = X _ni I(|X _ni | ≤ n ^1/q ) + n ^1/q I(X _ni > n ^1/q ) − n ^1/q I(X _ni < −n ^1/q ).

So that {Y ni |1 ≤ i ≤ n, n ≥ 1} is still an array of rowwise N D random variables by definition.

Next,

(6)

X n i=1

a ni + X ni I(|X ni | ≤ n ^1/q )

= X n i=1

a ni + (Y ni − EY ni )

− n ^1/q X n n=1

a ni + (I(X ni > n ^1/q ) − P (X ni > n ^1/q ))

+ n ^1/q X n n=1

a ni + (I(X ni < −n ^1/q ) − P (X ni < −n ^1/q ))

+ X n i=1

a ni + EX ni I(|X ni | ≤ n ^1/q )

= I 4 + I 5 + I 6 + I 7 (say).

As to I 4 , we consider two cases of (a) p ≥ 1 and (b) 0 < p < 1, and note that {a ni + (Y ni − EY ni ) | 1 ≤ i ≤ n, n ≥ 1} is still an array of rowwise N D random variables by definition and |a ni + (Y ni − EY ni )| ≤ 2n ^{−1/p + 1/q} = 2n ^−α .

(a) when p ≥ 1, note that

E|Y ni | ² ≤ E|X ni | ² I(|X ni | ≤ n ^1/q ) + n ^2/q P (|X ni | > n ^1/q )

≤ E|X| ² I(|X| ≤ n ^1/q ) + n ^2/q P (|X| > n ^1/q )

≤ 2E|X| ² < ∞

which implies that E|Y ni | ² < ∞ since E|X| ^2p < ∞ implies E|X| ² < ∞. Hence, by using Lemma 2.2 and Lemma 2.4 and taking t = 2 log n/², we get

I 4 = P Ã _n

X

i=1

a ni + (Y ni − EY ni ) > ²

!

≤ e ^{−²t Q}

^Ee

≤ e ^{−2 log n Q}

^e

≤ e ^{−2 log n Q}

^e

≤ e ^{−2 log n Q}

^e

≤ e ^{−2 log n} e ^{c(log n)}

^P

^(a

⁾

≤ e ^{−2 log n} e ^{c(log n)}

^·(

⁾

≤ ce ^{−c log n} −→ 0 as n → ∞.

Hence,

X ∞ n=1

P Ã _n

X

i=1

a ni + (Y ni − EY ni ) > ²

!

< ∞.

(b) when 0 < p < 1, taking t = n ^α /2, we get I 4 = P

à _n X

i=1

a ni + (Y ni − EY ni ) > ²

!

≤ e ^−²t e ^t

^P

^(a

⁾

^EY

≤ e ^−²t e ^cn

^P

ⁿ

^EX

^I(|X

^|≤n

⁾⁺ⁿ

^{P (|X}

^|>n

⁾

≤ e ^−²t e ^cn

ⁿ

^c(1+n

⁾

≤ e ^−cn

e ^cn

which is summable since α > 0 and 0 < p < q < 2p < 2 implies that 1−2/q < 0 and 1 − 2p/q < 0. Hence, by (a) and (b), for all 0 < p < 2,

(7)

X ∞ n=1

P Ã _n

X

i=1

a ni + (Y ni − EY ni ) > ²

!

< ∞.

As to I 5 , let Z ni = n ^1/q a ni + (I(X ni > n ^1/q )−P (X ni > n ^1/q )) and noticing that

|Z ni | = |n ^1/q a ni + (I(X ni > n ^1/q ) − P (X ni > n ^1/q ))| ≤ n ^1/q−1/p ≤ n ^−α and EZ ni 2 ≤ n ^2/q (a ni + ) ² P (X ni > n ^1/q ) + P (X ni > n ^1/q )

≤ cn ^2/q n ^−2/p E|X| ^2p /n ^2p/q , we get that

I 5 = P Ã _n

X

i=1

Z ni > ²

!

= P Ã

n ^α X n i=1

Z ni > n ^α ²

!

≤ e ^−²n

Ee ⁿ

^P

^Z

≤ e ^−²n

Y n i=1

e ⁿ

^E(Z

⁾

≤ e ^−²n

Y n i=1

e ^n(2/p−2/q) cn ^2/q−2/p E|X| ^2p /n ^2p/q

≤ e ^−²n

e ^cn(1−2p/q)

≤ e ^−²n

^+cn

which is summable since α > 0 and 1 − 2p/q < 0. Hence, (8)

X ∞ n=1

P Ã

n ^1/q X n i=1

a ni + (I(X ni > n ^1/q ) − P (X ni > n ^1/q )) > ²

!

< ∞.

(9) As to I 6 , the proof of I 6 is similar to I 5 .

Next, as to I 7 , we consider three cases of (c) p > 1/2, (d) p = 1/2, and (e) 0 < p < 1/2. Since EX ni = 0, it follows that

EX ni I(|X ni | ≤ n ^1/q ) = | − EX ni I(|X ni | > n ^1/q )|.

(c) when p > 1/2,

|I 7 | ≤ X n i=1

|a ni + EX ni I(|X ni | ≤ n ^1/q )|

≤ X n i=1

n ^−1/p E|X ni |I(|X ni | > n ^1/q )

≤ X n i=1

n ^−1/p (n ^1/q P (|X ni | > n ^1/q ) + Z _∞

n

P (|X ni | > t)dt)

≤ n ^1−1/p+1/q P (|X| > n ^1/q ) + n ^1−1/p Z _∞

n

P (|X| > t)dt

≤ n 1−1/p+1/q−2p/q E|X| ^2p + n 1−1/p+1/q−2p/q E|X| ^2p

≤ cn 1−1/p+1/q(1−2p)

which implies that X n i=1

a ni + EX ni I(|X ni | ≤ n ^1/q ) −→ 0 as n → ∞

since 1 − 1/p + 1/q(1 − 2p) < 0.

(d) when p = 1/2,

|I 7 | ≤ X n i=1

|a ni + EX ni I(|X ni | ≤ n ^1/q )|

≤ X n i=1

n ^−1/p E|X|I(|X| ≤ n ^1/q )

≤ n ^1−1/p E|X| −→ 0 as n → ∞.

(e) when 0 < p < 1/2, note that E|X| ^2p < ∞ implies that P (|X| > t) ≤ t ^−2t , where t ≥ A for some constant A. Hence, for n ^1/q ≥ A,

|I 7 | ≤ X n i=1

|a ni + EX ni I(|X ni | ≤ n ^1/q )|

≤ X n i=1

n ^−1/p E|X ni |I(|X ni | ≤ n ^1/q )

≤ n ^1−1/p E|X|I(|X| ≤ n ^1/q )

≤ n ^1−1/p Z _n

0

P (|X| > t)dt

= n ^1−1/p ÃZ _A

0

P (|X| > t)dt + Z _n

A

P (|X| > t)dt

!

≤ n ^1−1/p Ã

A + Z _n

A

P (|X| > t)dt

!

≤ n ^1−1/p µ

A + 1

1 − 2p n ^1/q(1−2p)

¶

≤ c 1

1 − 2p n 1−1/p + 1/q(1−2p)

which implies that X n i=1

a ni + EX ni I(|X ni | ≤ n ^1/q ) −→ 0 as n → ∞

since 1/q(1 − 2p) < 0 and 1 − 1/p < 0. Hence, by (c), (d), and (e), we get (10)

X ∞ n=1

P Ã _n

X

i=1

a ni + EX ni I(|X ni | ≤ n ^1/q )

!

< ∞.

Thus, by (7), (8), (9), and (10), we have

(11) I 1 =

X ∞ n=1

P ( X n i=1

a ni + X ni I(|X ni | ≤ n ^1/q ) ≥ ²/2) < ∞.

As to I 2 , we get that

P (a ni + X ni ≥ ²/2 for some i, 1 ≤ i ≤ n)

≤ P Ã _n

[

i=1

|a ni + X ni | > ²/2

!

≤ X n i=1

P (|a ni + X ni | > ²/2)

≤ X n i=1

P (n ^−1/p |X ni | > ²/2)

≤ X n i=1

P (|X| > n ^1/p ²/2)

≤ n ⁻¹ µ 2

²

¶ _2p E|X| ^2p

which implies that

n ⁻¹ E|X| ^2p −→ 0 as n → ∞.

Hence,

(12) I 2 = X ∞ n=1

P (a ni + X ni ≥ ²/2 for some i, 1 ≤ i ≤ n) < ∞.

Finally, we get that

P (X ni > n ^1/q for at least two values of 1 ≤ i ≤ n)

≤ P



 [

i6=j

X ni > n ^1/q , X nj > n ^1/q





≤ X

i6=j

P (|X ni | > n ^1/q , |X nj | > n ^1/q )

≤ X

i6=j

P (|X| > n ^1/q )P (|X| > n ^1/q )

≤ X

i6=j

E|X| ^2p (n ^−1/q ) ^2p E|X| ^2p (n ^−1/q ) ^2p

≤ cn(n − 1)n ^−4q/p

≤ cn ^{−2(2p/q−1)} which implies that

n ^{−2(2p/q−1)} −→ 0 as n → ∞ since 2p/q − 1 > 0. Hence,

(13) I 3 = X ∞ n=1

P (X ni > n ^1/q for at least two values of i, 1 ≤ i ≤ n) < ∞.

Thus, by (11), (12), and (13), we have X ∞

n=1

P ï ¯

¯ ¯

¯ X n i=1

a ⁺ _ni X ni

¯ ¯

¯ ¯

¯ > ²

!

< ∞ for all ² > 0.

By replacing X ni by (−X ni ) from (3) and noticing {a ⁺ _ni (−X ni ) | 1 ≤ i ≤ n, n ≥ 1} is still an array of rowwise N D random variables by definition, we also know that

X ∞ n=1

P Ã _n

X

n=1

a ⁺ _ni (−X _ni ) > ²

!

< ∞ for all ² > 0.

The proof is complete. ¤

Corollary 3.1. Let 0 < p < 2 and let {X ni | 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise N D random variables with EX ni = 0. Suppose that there is a random variables X such that sup P (|X ni | > x) ≤ P (|X| > x) for all i, n and x ≥ 0. Assume that {b ni |1 ≤ i ≤ n, n > 1|} is an array of constants satisfying lim n→∞ sup P _n

i=1 b ² _ni < ∞. If E|X| ^2p < ∞, then X n

i=1

b ni X ni / log n −→ 0 completely as n → ∞.

Proof. Let a ni = b ni / log n. Then we can obtain the result by Theorem 3.1.

The proof is complete. ¤

Theorem 3.2. Let 0 < p < 2 and let {X _ni |1 ≤ i ≤ n, n ≥ 1} be an array of rowwise N D random variables with EX ni = 0. Assume that {a ni | 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers satisfying max 1≤i≤n |a ni | = O(n ^−1/p ).

If |X ni | ≤ M , then X n i=1

a ni X ni −→ 0 completely as n → ∞.

Proof. Without loss of generality, we assume that 0 < a ⁺ _ni ≤ n ^−1/p for 1 ≤ i ≤ n, n ≥ 1.

As for the proof of Theorem 3.1, it suffices to show that X ∞

n=1

P Ã _n

X

i=1

a ⁺ _ni X ni > ²

!

< ∞ for all ² > 0.

We also know that {a ⁺ _ni X ni | 1 ≤ i ≤ n, n ≥ 1} is still an array of rowwise N D random variables by definition and |a ⁺ _ni X ni | ≤ n ^−1/p M and Ea ⁺ _ni X ni = 0.

Hence,

X ∞ n=1

P Ã _n

X

i=1

a ⁺ _ni X ni > ²

!

= X ∞ n=1

P Ã

e ^²n

^/M X n i=1

a ⁺ _ni X ni > e ^²

ⁿ

^/M

!

≤ X ∞ n=1

e ^−²

ⁿ

^/M Y n i=1

Ee ^²n

^/M (a ⁺ _ni X ni )

≤ X ∞ n=1

e ^−²

ⁿ

^/M Y n i=1

e ^(²n

^{/M )}

(a ⁺ _ni ) ² E(X ni ) ²

≤ c X ∞ n=1

e ^−²

ⁿ

^/M Y n i=1

e ^²

^/n

≤ c X ∞ n=1

e ^−²

ⁿ

^/M < ∞

since 0 < p < 2 and 1/p − 1/2 > 0. The proof is complete. ¤ Proposition 3.1. Let {X ni |1 ≤ i ≤ n, n ≥ 1} be an array of rowwise random variables. Suppose that there is a random variable X such that sup P (|X ni | >

x) ≤ P (|X| > x) for all i, n and x ≥ 0 and let E|X| ^2p < ∞ for some 0 < p < 1.

Assume that {a ni |1 ≤ i ≤ n, n ≥ 1} is an array of real numbers satisfying

1≤i≤n max |a ni | = O(n ^−1/p ) for some 0 < p < 1/2. If 0 < p < 1/2, then X n

i=1

a ni X ni −→ 0 completely as n → ∞.

Proof. Without loss of generality, we assume that 0 < a ⁺ _ni ≤ n ^−1/p for some 0 < p < 1/2 and 1 ≤ i ≤ n, n ≥ 1. As for the proof of Theorem 3.1, it suffices to show that

X ∞ n=1

P ï ¯

¯ ¯

¯ X n i=1

a ⁺ _ni X ni

¯ ¯

¯ ¯

¯ > ²

!

< ∞ for all ² > 0.

Then,

P ï ¯

¯ ¯

¯ X n i=1

a ⁺ _ni X ni

¯ ¯

¯ ¯

¯ > ²

!

≤ P ï ¯

¯ ¯

¯ X n i=1

a ⁺ _ni X ni I(|X ni | ≤ n ^1/p )

¯ ¯

¯ ¯

¯ > ²/2

!

+P ï ¯

¯ ¯

¯ X n i=1

a ⁺ _ni X ni I(|X ni | > n ^1/p )

¯ ¯

¯ ¯

¯ > ²/2

!

= I 8 + I 9 (say).

As to I 8 ,

I 8 = P ï ¯

¯ ¯

¯ X n i=1

a ⁺ _ni X ni I(|X ni | ≤ n ^1/p )

¯ ¯

¯ ¯

¯ > ²/2

!

≤ 2/² E

¯ ¯

¯ ¯

¯ X n

i=1

a ⁺ _ni X ni I(|X ni | ≤ n ^1/p )

¯ ¯

¯ ¯

¯

≤ 2/² n ^−1/p X n i=1

E|X|I(|X| ≤ n ^1/p )

≤ cn ^1−1/p Z _n

0

P (|X| > t)dt taking t = n ^1/p r

= cn Z ₁

0

P (|X| > n ^1/p r)dr

≤ cn ⁻¹ E|X| ^2p Z ₁

0

r ^−2p dr

which implies that

(14) n ⁻¹ E|X| ^2p

Z ₁

0

r ^−2p dr −→ 0 as n → ∞

since 0 < p < 1/2. As to I 9 ,

I 9 = P ï ¯

¯ ¯

¯ X n i=1

a ⁺ _ni X ni I(|X ni | > n ^1/p )

¯ ¯

¯ ¯

¯ > ²/2

!

≤ P Ã _n

X

i=1

n ^−1/p |X ni I(|X ni | > n ^1/p |) > ²/2

!

≤ P Ã _n

[

i=1

|X ni | > n ^1/p

!

(15)

≤ X n i=1

P (|X ni | > n ^1/p )

≤ nP (|X| ^p > n)

≤ n ⁻¹ E|X| ^2p −→ 0 as n → ∞.

Hence, by (14) and (15), P _∞

n=1 P (| P _n

i=1 a ⁺ _ni X ni | > ²) < ∞ for all ² > 0. The

proof is complete. ¤

Proposition 3.2. Let {X ni |1 ≤ i ≤ n, n ≥ 1} be an array of rowwise random variables. Suppose that there is a random variable X such that sup P (|X ni | >

x) ≤ P (|X| > x) for all i, n and x ≥ 0 and let E|X| ^2p < ∞ for some 0 < p < 1.

Assume that {a _ni |1 ≤ i ≤ n, n ≥ 1} is an array of real numbers satisfying

1≤i≤n max |a ni | = O(n ^−1/p ) for some 1/2 ≤ p < 1. If 1/2 ≤ p < 1, then X n

i=1

a ni X ni −→ 0 in probability.

Proof. It suffices to show that X n i=1

E|a ni X ni | → 0

which implies the weak law of large numbers. By sup P (|X ni | > x) ≤ P (|X| >

x), note that p ≥ 1/2 yield sup E|X ni | < E|X| < ∞. Hence, X n

i=1

E|a _ni X _ni | ≤ X n

i=1

n ^−1/p E|X _ni | ≤ n ^1−1/p E|X| −→ 0 as n → ∞

since 1/2 ≤ p < 1. Hence P _n

i=1 E|a ni X ni | → 0. The proof is complete. ¤ Acknowledgements. The author would like to thank to the referee for a detailed list of comments and suggestions which greatly improved the presen- tation of this paper, and this research was partially supported by Wonkwang University Grant in 2008.

References

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Jong-Il Baek

School of Mathematics and Informational Statistics and Basic Natural Science

Wonkwang University IkSan 570-749, Korea

E-mail address: [email protected] Hye-Young Seo

School of Mathematics and Informational Statistics and Basic Natural Science

Wonkwang University IkSan 570-749, Korea

E-mail address: [email protected] Gil-Hwan Lee

School of Mathematics and Informational Statistics and Basic Natural Science

Wonkwang University IkSan 570-749, Korea

E-mail address: [email protected] Jeong-Yeol Choi

School of Mathematics and Informational Statistics and Basic Natural Science

Wonkwang University IkSan 570-749, Korea

E-mail address: [email protected]