RANDOM VARIABLES

Vol. 22, No. 1 (1999) 171–177 S 0161-17129922171-0

© Electronic Publishing House

MARCINKIEWICZ-TYPE STRONG LAW OF LARGE NUMBERS FOR DOUBLE ARRAYS OF PAIRWISE INDEPENDENT

RANDOM VARIABLES

DUG HUN HONG and SEOK YOON HWANG (Received 31 May 1996)

Abstract.Let X_ij

be a double sequence of pairwise independent random variables.

If P

|Xmn| ≥t

≤P

|X| ≥t

for all nonnegative real numbers tandE|X|^p

log⁺|X|₃

<∞, for 1< p <2, then we prove that

_m

i=1_n j=1

X_ij−EX_ij

(mn)^1/p →0 a.s. asm∨n→ ∞. (0.1)

Under the weak condition ofE|X|^plog⁺|X|<∞, it converges to 0 inL¹. And the results can be generalized to anr-dimensional array of random variables under the conditions E|X|^p

log⁺|X|_r+1

<∞,E|X|^p

log⁺|X|_r−1

<∞, respectively, thus, extending Choi and Sung’s result [1] of the one-dimensional case.

Keywords and phrases. Strong law of large numbers, pairwise independent.

1991 Mathematics Subject Classification. 60F15.

1. Introduction. Etemadi [3] extended the classical law of large numbers for i.i.d.

random variables to the case where the random variables are pairwise i.i.d., i.e., if {Xn} is a sequence of pairwise i.i.d. random variables withE|X1|<∞, then

_n

i=1

Xi−EXi

n →0 a.s. (1.1)

In 1985, Choi and Sung [1] have shown that if {Xn}are pairwise independent and are dominated in distribution by a random variable Xwith E|X|^p

log⁺|X|₂

<∞,1<

p <2, then ⁿⁱ⁼¹_n^(X1/p^i−EXi) →0 a.s. In addition, ifE|X|^p<∞, then it converges to 0 in L¹.

For a double sequence Xij

of pairwise i.i.d. random variables, also Etemadi [3]

proved that ifE|X11|log⁺|X11|<∞, then _m

i=1_n

j=1

Xij−EXij

mn →0 a.s. asm∨n → ∞. (1.2)

Now, we are interested in the extension of Choi and Sung’s result of the one- dimensional case to a multi-dimensional array of pairwise independent random vari- ables, which is established in the next section.

2. Main results. Let Xij

be a double sequence of random variables and letX_ij= XijI

|Xij| ≤(ij)^1/p

, X_ij=XijI

|Xij|> (ij)^1/p

for 1< p <2. Throughout this paper,

c denotes an unimportant positive constant which is allowed to change and dkthe number of all divisors of integerk.

To prove the main theorem, we need the following lemmas.

Lemma2.1. Let Xij

be a double sequence of pairwise independent random vari- ables. If P

|Xmn| ≥t

≤P

|X| ≥t

for all nonnegative real numberst, then (a)

∞ i=1

∞ j=1

EX_ij²

(ij)^2/p ≤cE|X|^plog⁺|X|, (b)

∞ i=1

∞ j=1

EX_ij

(ij)^1/p ≤cE|X|^plog⁺|X| for1< p <2.

(2.1)

Proof. The estimation of E|X_ij|²is given by

EX_ij²≤ _(ij)^2/p

0 P Xij²≥t dt

≤ _(ij)^2/p

0 P |X|²≥t dt

= _(ij)2/p

0

P t≤ |X|²< (ij)^2/p

+P (ij)^2/p≤ |X|² dt

= _(ij)^1/p

0 x²dF(x)+(ij)^2/pP (ij)^2/p≤ |X|² ,

(2.2)

where F(x)is the distribution ofX. If we use the fact that _∞

k=i+1dk/k^2/p=O logi/

(i+1)^2/p−1

, we obtain ∞

i=1

∞ j=1

1 (ij)^2/p

_(ij)^1/p

0 x²dF(x)≤c ∞ k=1

dk

k^2/p _k^1/p

0 x²dF(x)

≤c ∞ i=0



 ^∞

k=i+1

dk

k^2/p



_(i+1)^1/p

i^1/p x²dF(x)

≤c ∞ i=0

logi (i+1)^2/p−1

_(i+1)^1/p

i^1/p x²dF(x)

≤cE|X|^plog⁺|X|<∞.

(2.3)

And

∞ i=1

∞ j=1

P

(ij)^2/p≤ |X|²

= ∞ k=1

dkP

k≤ |X|^p

= ∞ k=1



^k

j=1

dj



P

k≤ |X|^p< k+1

=c ∞ k=1

klogkP

k≤ |X|^p< k+1

≤cE|X|^plog⁺|X|<∞,

(2.4)

where we use the fact that_n

k=1dk=O(nlogn). It follows that ∞

i=1

∞ j=1

EX_ij²

(ij)^2/p <∞, which proves (a). (2.5) By the fact that _n

k=1dk/k^1/p =O

n^1−(1/p)logn

, we can obtain (b) by the same method.

The following lemma is a two parameter analog of [5, Lem. 3.6.1a].

Lemma2.2. Let Xij

be a double sequence of pairwise independent random vari- ables with EXij=0, and letSmn=_m

i=1_n

j=1Xij. Then

E



max

1≤i≤m 1≤j≤n

Sij



2

≤c(logm)²(logn)^2m

k=1

n l=1

EXkl². (2.6) Proof. For m=1 and n=1, the inequality is trivial. If m >1, let sbe an integer such that 2^s−1< m≤2^s. And if n >1, let tbe an integer such that 2^t−1< n≤2^t. We can assume thatm, n >1. We assign Xijto the point (i,j)of integer in (0,2^s]×(0,2^t] (if m < i≤2^s or n < j≤2^t, setXij=0). Divide the interval(0,2^s]into (0,2^s−1]and (2^s−1,2^s], each of these two intervals into two halves, and so on. Then the elements of theith partition are of length 2^s−i, i=0,...,s. Also, divide the interval(0,2^t]in the same way. Then we obtain the(i,j)th partition Pij of(0,2^s]×(0,2^t]by theith partition of (0,2^s] and the jth partition of (0,2^t]. Every rectangle (0,i]×(0,j] is the sum of at most (s+1)(t+1)disjoint subrectangles each of which belongs to a different partition. We can write Sij =_s

k=0

_t

l=0Ykl;ij, where Ykl;ij is the sum of all r.v.’s belonging to the rectangle (a,b]×(c,d], b−a=2^kand d−c=2^l, which may or may not be a summand of (0,i]×(0,j]so that some Ykl;ij may vanish. Let Tij= ₂i

k=1

₂j

l=1|Ykl|², where Yklis the sum of all r.v.’s which belong to the(k,l)-element of Pij. If we put T=_s

i=0_t

j=0Tij, by the elementary Schwarz inequality, we obtain Sij²≤(s+1)(t+1)

s k=0

t l=0

Ykl;ij²≤(s+1)(t+1)T . (2.7) SinceETij≤_m

k=1_n

l=1EXkl², ET≤(s+1)(t+1)_m

k=1_n

l=1EXkl². It follows that E



max

1≤i≤m 1≤j≤n

Sij²



≤(s+1)²(t+1)² m k=1

n l=1

EXkl²

≤c(logm)²(logn)^2m

k=1

n l=1

EXkl².

(2.8)

Theorem2.3. Let Xij

be a double sequence of pairwise independent random vari- ables. If P

|Xmn| ≥t

≤P

|X| ≥t

for all nonnegative real numbers tandE|X|^p log⁺

|X|3<∞, for 1< p <2, then

m∨nlim→∞

_m

i=1_n

j=1

Xij−EXij

(mn)^1/p =0 a.s. (2.9)

Proof. We denote bySmn=_m

i=1_n

j=1Xij, Smn =_m

i=1_n

j=1X_ij. Then we obtain the inequalities

∞ i=1

∞ j=1

P

Xij=X_ij

= ∞ k=1

dkP

|X11|> k^1/p

≤ ∞ k=1

dkP

|X|> k^1/p

=^∞

i=1



ⁱ

k=1

dk



_(i+1)^1/p

i^1/p dF(x)

≤c ∞ i=1

ilogi

_(i+1)1/p

i^1/p dF(x)

≤cE|X|^plog⁺|X|<∞,

(2.10)

Hence, by the Borel-Cantelli lemma, _m

i=1

_n

j=1

Xij−X_ij

(mn)^1/p →0 a.s. (2.11)

Now, we use Chebyshev’s inequality and Lemma 2.1 to obtain ∞

k=1

∞ l=1

P

S₂k2^l−ES₂k2^l

2^k2^l_1/p > "

≤c ∞ k=1

∞ l=1

VarS₂k2^l

2^k2^l_2/p

=c ∞ k=1

∞ l=1

1 2^k2^l2/p

2^k

i=1 2^l

j=1

VarX_ij

≤c ∞ i=1

∞ j=1

EX_ij² (ij)^2/p

≤cE|X|^plog⁺|X|^p<∞,

(2.12)

which follows easily by summation by parts. It follows that S₂k2^l−ES₂k2^l

2^k2^l_1/p →0 a.s. (2.13)

And let

Tkl= max

2^k≤m<2^k+1 2^l≤n<2^l+1

S₂^∗k2^l

2^k2^l1/p− S_mn^∗ (mn)^1/p

≤ S₂^∗k2^l

2^k2^l_1/p+ max

2^k≤m<2^k+1 2^l≤n<2^l+1

Smn^∗ (mn)^1/p,

(2.14)

whereSmn^∗ =Smn −ESmn .

By using Lemma 2.2, we obtain, for any" >0,

∞ k=0

∞ l=0

P





 max

2^k≤m≤2^k+1 2^l≤n≤2^l+1

S_mn^∗ (mn)^1/p ≥"

2





≤c^∞

k=0

∞ l=0

1

2^k2^l_2/pE





 max

2^k≤m≤2^k+1 2^l≤n≤2^l+1

S_mn^∗







2

≤c ∞ k=0

∞ l=0

(k+1)²(l+1)² 2^k+12^l+1_2/p

2^k+1

i=1 2^l+1

j=1

EX_ij²

≤c ∞ i=1

∞ j=1

(log₂ij)²

(ij)^2/p EX_ij²,

(2.15)

where the last inequality follows easily be summation by parts. But ∞

i=1

∞ j=1

(log2ij)²

(ij)^2/p EX_ij²≤ ∞ k=1

dk(log2k)² k^2/p

_k^1/p

0 x²dF(x)

≤ ∞ i=0



 ^∞

k=i+1

dk(log₂k)² k^2/p



_(i+1)^1/p

i^1/p x²dF(x)

≤c ∞ i=0

i^1−(2/p)(logi)³

_(i+1)^1/p

i^1/p x²dF(x)

≤cE|X|^p

log⁺|X|^p3

<∞,

(2.16)

where we use _∞

k=1dk(log2k)²

k^2/p =O _i^(logi)(2/p)−1³

which follows by summation by parts.

Hence, (2.13), (2.15), and (2.16) give us S_mn −ES_mn

(mn)^1/p →0 a.s. (2.17)

Combining (2.11) and (2.17), we get Smn−ES_mn

(mn)^1/p →0 a.s. (2.18)

Since

Smn−ESmn

(mn)^1/p =Smn−ES_mn (mn)^1/p −

_m

i=1_n

j=1EX_ij

(mn)^1/p , (2.19)

it remains to prove that the second term of the right-hand side converges to 0 a.s. By Lemma 2.1(b), we obtain

∞ k=1

∞ l=1

₂^k

i=1

₂^l

j=1EX_ij 2^k2^l1/p ≤c

∞ i,j=1

EX_ij (ij)^1/p

≤cE|X|^plog⁺|x|<∞,

(2.20)

from which, it follows that

k∨llim→∞

₂^k

i=1₂^l

j=1EX_ij

2^k2^l_1/p =0. (2.21)

But since

T_kl = max

2^k≤m≤2^k+1 2^l≤n≤2^l+1

_m

i=1

_n

j=1EX_ij (mn)^1/p −

₂k i=1

₂l

j=1EX_ij 2^k2^l_1/p

≤ c

2^k+12^l+11/p 2^k+1

i=1 2^l+1

j=1

EX_ij,

(2.22)

T_kl converges to 0 which implies that, by (2.21), _m

i=1

_n

j=1EX_ij

(mn)^1/p →0. (2.23)

This completes the proof.

Corollary2.4. Let Xij

be a double sequence of pairwise i.i.d. random variables with E|X11|^p

log⁺|X11|₃

<∞, for1< p <2. Then

m∨nlim→∞

_m

i=1_n

j=1

Xij−EXij

(mn)^1/p =0 a.s. (2.24)

Remark. The generalization tor-dimensional arrays of random variables can be obtained easily under the conditionE|X|^p

log⁺|X|r+1<∞.

Theorem2.5. Let Xij

be a double sequence of pairwise independent random vari- ables. If P

|Xij| ≥t

≤P

|X| ≥t

for all nonnegative real numbers tand E|X|^plog⁺

|X|<∞,1< p <2, then _m

i=1_n

j=1

Xij−EXij

(mn)^1/p →0 inL¹asm∨n → ∞. (2.25) Proof. Since

Xij

are pairwise independent,

X_ij −EX_ij

are orthogonal which implies that

E

_m

i=1_n

j=1

X_ij−EX_ij (mn)^1/p

2

≤ _m

i=1_n

j=1EX_ij²

(mn)^2/p . (2.26)

Since

E

_m

i=1_n

j=1

Xij−EXij (mn)^1/p

≤E

_m

i=1_n

j=1

X_ij −EX_ij (mn)^1/p

+2

_m

i=1

_n

j=1EX_ij (mn)^1/p ,

(2.27)

it suffices to show that _m

i=1_n

j=1E|X_ij|²

/(mn)^2/p converges to 0 asm∨n →0.

But this can be shown by a method similar to that used in the proof of (2.23) in Theorem 2.3.

Corollary2.6. Let Xij

be a double sequence of pairwise i.i.d. random variable withE|X11|^plog⁺|X11|<∞, for1< p <2. Then

_m

i=1_n

j=1

Xij−EXij

(mn)^1/p →0 inL¹asm∨n → ∞. (2.28)

Remark. The generalization tor-dimensional arrays of random variables can be obtained under the conditionE|X|^p

log⁺|X|r+1<∞.

Acknowledgement. This research was supported by the Catholic University of Taegu-Hyosung.

References

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MR 87f:60044. Zbl 585.60043.

[2] K. L. Chung,A course in probability theory, second ed., Probability and Mathematical Sta- tistics, vol. 21, Academic Press, New York, 1974. MR 49 11579. Zbl 345.60003.

[3] N. Etemadi,An elementary proof of the strong law of large numbers, Z. Wahrsch. Verw.

Gebiete55(1981), no. 1, 119–122. MR 82b:60027. Zbl 448.60024.

[4] G. H. Hardy and E. M. Wright,An introduction to the theory of numbers, fourth ed., Claren- don Press, Oxford, 1960. Zbl 086.25803.

[5] M. Loeve,Probability theory. I, fourth ed., Graduate Texts in Mathematics, vol. 45, Springer- Verlag, New York, 1977. MR 58 31324a. Zbl 359.60001.

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Hong: School of Mechanical and Automotive Engineering, Catholic University of Taegu-Hyosung, Kyungbuk712-702, South Korea

Hwang: Department of Mathematics, Taegu University, Kyungbuk713-714, South Ko- rea

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