ON THE LP -CONVERGENCE FOR MULTIDIMENSIONAL
ARRAYS OF RANDOM VARIABLES
LE VAN THANH
Received 14 August 2004 and in revised form 4 March 2005
For a d-dimensional array of random variables {Xn , n ∈ Zd+ } such that {|Xn | p , n ∈ Zd+ } is
uniformly integrable for some 0 < p < 2, the L p -convergence is established for the sums
(1/ |n|1/ p )( j ≺n (X j − a j )), where a j = 0 if 0 < p < 1, and a j = EX j if 1 ≤ p < 2.
1. Introduction
Let Zd+ , where d is an integer, denote the positive integer d-dimensional lattice points.
The notation m ≺ n, where m = (m1 ,m2 ,...,md ) and n = (n1 ,n2 ,...,nd ) ∈ Zd+ , means that
mi ≤ ni , 1 ≤ i ≤ d, |n| is used for di=1 ni .
Gut [2] proved that if {X,Xn , n ∈ Zd+ } is a d-dimensional array of i.i.d. random variables with E|X | p < ∞ (0 < p < 2) and EX = 0 if 1 ≤ p < 2, then
j ≺n X j
|n|1/ p
−→ 0 in L p
as min ni −→ ∞,
1≤i≤d
(1.1)
where (n1 ,n2 ,...,nd ) = n ∈ Zd+ .
In 1999, Hong and Hwang [3] proved that if {Xmn , m ≥ 1, n ≥ 1} is a double array of
pairwise independent random variables such that
P Xmn > t ≤ P |X | > t ,
t ≥ 0, m ≥ 1, n ≥ 1,
(1.2)
where X is a random variable, then the condition E(|X | p log+ |X |) < ∞ (1 < p < 2) implies
that
m n k =1
Xkl − EXkl
−→ 0 in L1
(mn)1/ p
l=1
as max{m,n} −→ ∞.
(1.3)
In this note, we provide conditions for (1/ |n|1/ p )( j ≺n (X j − a j )) → 0 in L p as |n| → ∞,
where n ∈ Zd+ , j ∈ Zd+ , a j = 0 if 0 < p < 1, and a j = EX j if 1 ≤ p < 2.
Copyright © 2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:8 (2005) 1317–1320
DOI: 10.1155/IJMMS.2005.1317
On the LP -convergence
1318
2. Result
Theorem 2.1. Let {Xn , n ∈ Zd+ } be a d-dimensional array of random variables such that
{|Xn | p , n ∈ Zd+ } is uniformly integrable for some 0 < p < 2. Assume that {Xn , n ∈ Zd+ } is
pairwise independent if p = 1 and {Xn , n ∈ Zd+ } is independent if 1 < p < 2. Then,
Xj − aj
−→ 0 in L p as |n| −→ ∞,
|n|1/ p
j ≺n
(2.1)
where a j = 0 if 0 < p < 1, and a j = EX j if 1 ≤ p < 2.
Proof. For arbitrary > 0, there exists M > 0 such that
p E Xn I Xn > M < ∀n ∈ Zd+ .
(2.2)
Set
Xn = Xn I Xn ≤ M ,
n ∈ Zd+ ,
X = Xn I Xn > M ,
(2.3)
n ∈ Zd+ .
n
For all n ∈ Zd+ ,
p
p
EXn − EXn ≤ 4EXn < 4.
(2.4)
If 0 < p < 1, then
p
p
p
p p
Xj
≤ E
Xj + E
Xj ≤ E
Xj + EX E
j
j ≺n
j ≺n
p
≤ |n|M + |n|
j ≺n
j ≺n
j ≺n
(2.5)
by (2.2) .
The conclusion (2.1) follows from (2.5).
If p = 1 and {Xn , n ∈ Zd+ } is pairwise independent, then
E
j ≺n
X j − EX j ≤ E
j ≺n
X j − EX j +
EX − EX j ≺n
j
j
 2 1/2
X
− EX j  +
EX ≤ E j
j − EX j
j ≺n
j ≺n
by the Jensen inequality see [1, page 103]
≤
j ≺n
2
E X j − EX j
1/2
+ 4|n|
by (2.4)
1/2
≤ |n|M 2
+ 4|n|
2
2 2
since E X j − EX j = E X j − EX j ≤ M 2 , j ∈ Zd+
as |n| −→ ∞.
= o |n|
(2.6)
Le Van Thanh 1319
If 1 < p < 2 and {Xn , n ∈ Zd+ } is independent, then
 p
p
p

X
− EX j ≤ 2 p−1 E
X
− EX j + E
X
− EX E
j
j
j
j
j ≺n
j ≺n
j ≺n


2  p/2
 
p 
≤ 2 p−1 E
X j − EX j + 2 EX 
j − EX j
j ≺n
j ≺n
by the Jensen inequality [1] and the von Bahr-Esseen
inequality [4]
≤ 2 p −1
j ≺n
E X j − EX j
2
(2.7)
p/2
+ 2 p+2 |n|
(by (2.4))
p/2
≤ 2 p−1 |n|M 2
+ 2 p+2 |n|
2
2 2
since E X j − EX j = E X j − EX j ≤ M 2 , j ∈ Zd+
as |n| −→ ∞,
= o |n|
again establishing (2.1).
Note that if {X,Xn , n ∈ Zd+ } are random variables such that E|X | p < ∞ (p > 0) and
supn∈Zd+ P {|Xn | > t } ≤ P {|X | > t } for all t ≥ 0, then {|Xn | p , n ∈ Zd+ } is uniformly integrable. The following corollary follows immediately from Theorem 2.1.
Corollary 2.2. Let {X,Xn , n ∈ Zd+ } be random variables such that E|X | p < ∞ for some
0 < p < 2, and supn∈Zd+ P {|Xn | > t } ≤ P {|X | > t } for all t ≥ 0. Assume that {Xn , n ∈ Zd+ } is
pairwise independent if p = 1 and {Xn , n ∈ Zd+ } is independent if 1 < p < 2. Then,
Xj − aj
−→ 0 in L p
|n|1/ p
j ≺n
as |n| −→ ∞,
(2.8)
where a j = 0 if 0 < p < 1, and a j = EX j if 1 ≤ p < 2.
Acknowledgment
The author is grateful to Professor Andrew Rosalsky (University of Florida, USA) for the
reference [2].
References
[1]
[2]
Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales,
3rd ed., Springer Texts in Statistics, Springer-Verlag, New York, 1997.
A. Gut, Marcinkiewicz laws and convergence rates in the law of large numbers for random variables
with multidimensional indices, Ann. Probab. 6 (1978), no. 3, 469–482.
1320
[3]
[4]
On the LP -convergence
D. H. Hong and S. Y. Hwang, Marcinkiewicz-type strong law of large numbers for double arrays of
pairwise independent random variables, Int. J. Math. Math. Sci. 22 (1999), no. 1, 171–177.
B. von Bahr and C. Esseen, Inequalities for the rth absolute moment of a sum of random variables,
1 ≤ r ≤ 2, Ann. Statist. 36 (1965), 299–303.
Le Van Thanh: Department of Mathematics, Vinh University, Nghe An 42118, Vietnam
E-mail address: lvthanhvinh@yahoo.com