Statistics & Probability Letters 53 (2001) 259 – 267
A weak convergence for negatively associated #elds
Li-Xin Zhang ∗ , Jiwei Wen
Department of Mathematics, Zhejiang University, Xixi Campus, Zhejiang, Hangzhou 310028, People’s Republic of China
Received August 1999; received in revised form October 2000
Abstract
The aim of this paper is to investigate the weak convergence for negatively associated random #elds. To this end, we
obtain some moment inequalities for the maximum of partial sums for a negatively associated random #eld, which also
c 2001 Elsevier Science B.V. All rights reserved
are of independent interest. MSC: primary 60F05; secondary 62E20
Keywords: Negative association; Random #eld; Weak convergence; Maximal inequalities
1. Introduction and main results
Let d be a positive integer, and let {Xn ; n ∈ Nd } be a #eld of random variables de#ned on a probability
space (; U; P). The #eld {Xn ; n ∈ Nd } is called negatively associated (NA), if for every pair of disjoint
subsets S; T of Nd and any pair of coordinatewise increasing functions f(Xi ; i ∈ S); g(Xj ; j ∈ T ) with
Ef2 (Xi ; i ∈ S) ¡ ∞ and Eg2 (Xj ; j ∈ T ) ¡ ∞, it holds that
cov(f(Xi ; i ∈ S); g(Xj ; j ∈ T )) 6 0:
The concept of NA was introduced by Joag-Dev and Proschan (1983). As pointed out and proved by Joag-Dev
and Proschan (1983), a number of well-known multivariate distributions possess the NA property, such as
multinomial distribution, multivariate hypergeometric distribution, Dirichlet distribution, negatively correlated
normal distribution, permutation distribution, and joint distribution of ranks. Because of their wide applications
in multivariate statistical analysis and reliability theory, the concept of negatively associated random variables
has received extensive attention recently. We refer to Joag-Dev and Proschan (1983) for fundamental properties. In the case of d = 1, we refer to Newman (1984) for the central limit theorem, Matula (1992) for
the three series theorem, Roussas (1996) for the HoeBding inequality, Shao (2000) for the Rosenthal-type
Research supported by National Natural Science Foundation of China (No. 10071072).
∗
Corresponding author.
E-mail addresses: lxzhang@mail.hz.zj.cn (L.-X. Zhang), wenjiwei@263.net (J. Wen).
c 2001 Elsevier Science B.V. All rights reserved
0167-7152/01/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 0 2 1 - 9
260
L.-X. Zhang, J. Wen / Statistics & Probability Letters 53 (2001) 259 – 267
inequality and the Kolmogorov exponential inequality, Shao and Su (1999) for the law of the iterated logarithm, and Su et al. (1997) for a moment inequality and weak convergence. In the case of d ¿ 2, Roussas
(1994) studied the central limit theorems for weak stationary NA random #elds, and Zhang (2000) obtained
the weak convergence. But the 2 + th moment is assumed to be #nite in both of these two papers. The aim
of this paper is to investigate the weak convergence for a NA #eld with only #nite second moment. To state
our results, we need some notation #rst.
d
Zd ; Nd denote d-dimensional lattices of integers, positive integers, respectively. Rd = (−∞; ∞)d ; R+
=
d
[0; ∞)d . “6” denotes the coordinatewise partial order on Zd ; Nd ; Rd and R+
: e = (1; : : : ; 1) ∈ Nd . For n ∈
d
; ∈ R+ , let ts = (t1 s1 ; : : : ; td sd ), [t] = ([t1 ]; : : : ; [td ]), |n| = n1 · · · nd ; t + s = (t1 + s1 ; : : : ; td + sd );
Nd ; t; s ∈ R+
t = (t1 ; : : : ; td ) and t + = (t1 + ; : : : ; td + ).
A #eld of random variables {Xn ; n ∈ N d } will be called (strong) stationary, if for any ∀p ∈ Nd ; {Xn+p ; n ∈
Nd } has the same joint distribution as {Xn ; n ∈ N d }, it is called identically distributed, if forany n; Xn has
the same distribution as Xe , and, it is called “centered”, if E Xn = 0 for all n ∈ Nd . Let Sn = k6n Xk be the
partial sums of {Xn ; n ∈ Nd }. Inspired by Peligrad (1998), we obtain the following results.
Theorem 1. Let {Xn ; n ∈ Nd } be a ;eld of stationary centered NA random variables with 0 ¡ E Xe2 ¡ ∞.
De;ne
S[nt]
Wn (t) = ;
|n|
t ∈ [0; 1]d :
(1.1)
n→∞
(1.2)
Then
ESn2
→ 2 ¡ ∞;
|n|
and
Wn ⇒ W;
n → ∞;
(1.3)
d
in the space D[0; 1]d endowed with the Skorohod topology; where {W (t); t ∈ R+
} is a d-dimensional standard
d
Wiener process; n → ∞ denotes n1 → ∞; : : : ; nd → ∞. If {Xn ; n ∈ Z } is the extension of {Xn ; n ∈ N d };
then
2 = var Xe +
cov(Xe ; Xp ):
(1.4)
p = e
p ∈ Zd
Remark. Zhang (1999, 2000) introduce the concept of − -mixing. A #eld of random variables {Xn ; n ∈ Nd }
is called − -mixing, if
− (s) = sup{− (S; T ): S; T ∈ Nd ; dist(S; T ) ¿ s} → 0
where
− (S; T ) = 0 ∨ sup
(s → ∞);
cov{f(Xi ; i ∈ S); g(Xj ; j ∈ T )}
: f; g ∈ F ;
var{f(Xi ; i ∈ S)} · var{g(Xj ; j ∈ T )}
F = {f = f(x1 ; : : : ; xp ): f is coordinatewise increasing; p ¿ 1}:
A NA #eld is obviously − -mixing with − (1) = 0. − -mixing de#nes a strictly larger class of random
variables than does negative association. For more details, one can refer to Zhang (1999, 2000). One can
show that Theorem 1 is also true for a − -mixing #eld.
L.-X. Zhang, J. Wen / Statistics & Probability Letters 53 (2001) 259 – 267
261
2. Proof of Theorem 1
First, we state a lemma required in the proof.
Lemma 1. Suppose that {X; Xn ; n ∈ Nd } is a ;eld of identically distributed NA random variables with
E X = 0 and E X 2 ¡ ∞. Let Mn = maxk6n |Sk |. Then
Mn2
d
are uniformly integrable:
;n ∈ N
|n| ∨ EMn2
Proof. For k 6 n, let
XO k; n = (− |n|) ∨ (Xk ∧ |n|);
Xk; n = XO k; n − E XO k; n ;
Sk; n =
Xm;
n;
Xk;n = Xk − Xk; n ;
Sk; n =
m6k
Xm;
n;
m6k
Mn = max|Sk; n |;
k6n
Mn = max|Sk; n |:
k6n
Then {Xk; n ; k 6 n} and {Xk;n ; k 6 n} are both #elds of centered NA random variables. It follows from
Lemma A.2 in the Appendix that
2
2
2
E(Mn ) 6 C (E Mn ) +
E(Xk; n )
6C

 
k6n
2
E|Xk;n |
k6n


+ |n|E X 2 I {|X | ¿ |n|}

6 C{(|n|E|X |I {|X | ¿ |n|})2 + |n|E X 2 I {|X | ¿ |n|}}
6 C|n|E X 2 I {|X | ¿ |n|} + C|n|(E X 2 I {|X | ¿ |n|})2
and
E(Mn )4 6 C{(E Mn2 )2 + (|n|E X 2 I {|X | 6
|n|})2 + |n|E X 4 I {|X | 6 |n|}}
6 C{(E Mn2 )2 + (E Mn2 )2 + |n|2 (E X 2 )2 + |n|2 E X 2 }
6 C{(E Mn2 )2 + |n|2 } 6 C(|n| ∨ E Mn2 )2 :
Thus
E Mn2 I {Mn2 ¿ #(|n| ∨ E Mn2 )}
|n| ∨ E Mn2
64
E(Mn )2
E(Mn )2
+4
I
|n| ∨ E Mn2
|n|
(Mn )2 ¿
#
(|n| ∨ E Mn2 )
2
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L.-X. Zhang, J. Wen / Statistics & Probability Letters 53 (2001) 259 – 267
6 CE X 2 I {|X | ¿
|n|} + C(E X 2 I {|X | ¿ |n|})2 + 16
6 CE X 2 I {|X | ¿
C
|n|} + C(E X 2 I {|X | ¿ |n|})2 + → 0;
#
E(Mn )4
#(|n| ∨ E Mn2 )2
as n → ∞; # → ∞, which completes the proof.
Now we turn to the proof of Theorem 1. First, we have the following central limit theorem.
Proposition 1. Let {Xn ; n ∈ Nd } be a ;eld of stationary centered NA random variables with 0 ¡ E Xe2 ¡ ∞.
Then
ESn2
→ 2
|n|
and
S D
n →N (0; 2 );
|n|
where 2 is de;ned as in (1:4).
Proof. The proof is similar to Newman and Wright (1981) and so is omitted.
By the central limit theorem, to prove Theorem 1 it is suPcient to show that {Wn (t); t ∈ [0; 1]d } is tight.
The following proposition gives us the required tightness.
Proposition 2. Let {Xn ; n ∈ Nd } be a ;eld of stationary centered NA random variables with 0 ¡ E Xe2 ¡ ∞.
Then there exists a positive constant K; depending only on d; such that
lim sup|n|−1 E max Sk2 6 KE Xe2
(2.1)
k6n
n→∞
and {Wn (t); t ∈ [0; 1]d } is tight in the space D[0; 1]d endowed with the Skorohod topology.
Proof. Let
an =
max E Mk2 =|k|:
k6n
Then, an is coordinatewise increasing. De#ne the random functions Vn (·) by
S[nt]
;
Vn (t) = |n|an
t ∈ [0; 1]d :
First, we prove that {Vn (·)} is tight. Let !(Vn ; ) = supt−s¡ |Vn (t) − Vn (s)|, where · is the Euclidean
norm in Rd . It is suPcient to prove that
lim lim supP(!(Vn ; ) ¿ () = 0;
→0
n
∀( ¿ 0:
(2.2)
L.-X. Zhang, J. Wen / Statistics & Probability Letters 53 (2001) 259 – 267
263
Noting that
!(Vn ; ) 6
6
d
sup |Vn (t1 ; : : : ; tp−1 ; sp ; : : : ; sd ) − Vn (t1 ; : : : ; tp−1 ; tp ; sp+1 ; : : : ; sd )|
p=1 t−s¡
d
sup
d
p=1 t∈[0;1]
|tp −sp |¡
|Vn (t1 ; : : : ; tp−1 ; tp ; : : : ; td ) − Vn (t1 ; : : : ; tp−1 ; sp ; tp+1 ; : : : ; td )|;
we only need to show that for each p = 1; : : : ; d and ∀( ¿ 0,
sup |Vn (t) − Vn (t1 ; : : : ; tp−1 ; sp ; tp+1 ; : : : ; td )| ¿ (
lim lim sup P
= 0:
t ∈ [0;1]d
→0
n
|tp −sp |¡
By the symmetry in coordinates, it is suPcient to show this result for the case of p = d. Similar to Theorem
8:4 of Billingsley (1968) and its proof, it is suPcient to prove that
sup
|Vn (t;O td ) − Vn (t;O i)| ¿ (
1
lim lim sup P
= 0; ∀( ¿ 0;
tO
→0
td ∈[i;(i+1)]
n
where tO = (t1 ; : : : ; td−1 ) for t = (t1 ; : : : ; td ), and in the follows nO = (n1 ; : : : ; nd−1 ) and kO = (k1 ; : : : ; kd−1 ) are
de#ned similarly. By noting that
1
sup|Vn (t;O td ) − Vn (t;O i)| 6 max|Sk;O [nd td ] − Sk;O [nd i] |
O nO
|n|an k6
tO
and that {Xn } is stationary, we need only prove that
1
lim lim sup P max max|Sk;O i | ¿ ( |n|an = 0:
O nO
→0
i6nd k6
n
By the Chebyshev inequality and since {an } is coordinatewise increasing, we have
2
Mn;O [nd ]
1
1 E Mn;O [nd ]
(
:
I ¿√
P max max|Sk;O i | ¿ ( |n|an 6
O nO
i6nd k6
(2 |n|a2n
|n|[n
O d ]an;O [nd ]
(2.3)
(2.4)
Since (|n|E Xe2 ) ∨ Mn2 6 |n|a2n , by Lemma 1, we have that
2
Mn
d
are uniformly integrable:
;
n
∈
N
|n|a2n
Then, (2.3) follows from (2:4).
Next, we prove (2.1). Choose K = E supt∈[0; 1]d W 2 (t). We prove (2.1) by induction on d. We shall prove
that (2.1) holds for d = 1 and that it holds for d whenever it holds for d − 1. Suppose that
lim sup|n|−1 E maxSk2 ¿ (K + ()E Xe2 :
k6n
n→∞
(2.5)
Then
O Xe2 := lim a2n ¿ (K + ()E Xe2 :
KE
n→∞
By the central limit theorem, we can show easily that the #nite dimensional distributions of {Wn (t); t ∈ [0; 1]d }
converge to those of {W (t); t ∈ [0; 1]d }. Noting that
1
O Xe2 ;
Vn (t) = Wn (t); a2n → KE
an
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L.-X. Zhang, J. Wen / Statistics & Probability Letters 53 (2001) 259 – 267
we conclude that the #nite dimensional distributions of {Vn (t); t ∈[0; 1]d } converge to those of {W (t)=
d
O Xe2 ;
KE
t ∈ [0; 1] }. Since {Vn (·)} is tight, it follows that
W (·)
:
Vn (·) ⇒ O Xe2
KE
Thus
|W (t)|
Mn
= sup |Vn (t)| ⇒ sup :
2
|n|an t∈[0;1]d
t∈[0;1]d
O Xe2
KE
Let bn = E Mn2 =|n|. Since {Mn2 =|n|a2n ; n ∈ Nd } are uniformly integrable, we have
bn
E Mn2
2
2 · K
2
:
=
→
E
sup
W
(t)
=
2
2
O Xe2 t∈[0;1]d
an
|n|an
KE
E Xe2 · KO
(2.6)
If KO = + ∞, then for any (0 ¡ ¡ 1), there exists ñ = (ñ1 ; : : : ; ñd ) such that
bn
¡ ;
a2n
∀n ¿ ñ:
Let m = (m1 ; : : : ; md ). When d = 1, it is obvious that for m 6 ñ,
E Mm2 6 m2 E X12 6 ñ · mE X12 ;
i.e., bm 6 C ñ. When d ¿ 2, by the hypothesis of induction, (2.1) holds for d − 1. So, by Lemma A.1 in the
Appendix, we also have for md 6 ñd ,
2
O
E Mm2 6 md max E max|Sk;2O kd | 6 max Ckd · md |m|E|S
e;O kd |
kd 6md
O mO
k6
kd 6md
6 max Ckd · md |m|m
O d 6 Cmd |m| 6 C|ñ| · |m|;
kd 6md
i.e., bm 6 C|ñ|, where C depends on d and ñd . Similarly, for each p = 1; : : : ; d, bm 6 C|ñ| if mp 6 ñp . It
follows that for n ¿ ñ,
1=
a2n
bm
bm
maxm6n bm
=
6 max max
+ max 2
ñ6m6n am
p6d m6n; mp 6ñp a2n
a2n
a2n
6
C|ñ|
+ → ;
a2n
(n → ∞):
This is a contradiction. If 0 ¡ KO ¡ ∞, then by (2:6) and noting that 2 6 E Xe2 , we obtain
E Mn2
2 K
6 K;
→
|n|
E Xe2
which is a contradiction to our assumption in (2.5), and so (2.1) is proved.
Finally, note that !(Wn ; ) = an · !(Vn ; ) and an 6 C. By (2.2), we have
lim lim supP(!(Wn ; ) ¿ () = 0;
→0
n
Thus, {Wn (t); t ∈ [0; 1]d } is tight.
∀( ¿ 0:
L.-X. Zhang, J. Wen / Statistics & Probability Letters 53 (2001) 259 – 267
265
Appendix
Lemma A.1. Let {Xn ; n ∈ Nd } be a ;eld of centered NA random variables. Then for all p ¿ 2; there
exists a positive constant C = Cp such that


p=2

 p
E Xk2
+
E|Xk |p ; ∀A ⊂ Nd :
E Xk 6 C


k∈A
k∈A
k∈A
Proof. See Zhang (1999) or the proof of Theorem 2 of Zhang (1998).
Lemma A.2. Let {Xn ; n ∈ Nd } be a ;eld of centered NA random variables. Then for all p ¿ 2; there
exists a positive constant C = Cp such that


p=2
p 

Emax|Sk |p 6 C
Emax|Sk | +
E Xk2
+
E|Xk |p ; ∀n ∈ Nd :

 k6n
k6n
k6n
k6n
Proof. Let Mn = maxk6n |Sk |. By Lemma A.1, it is suPcient to show that for all q ¿ 1, there exists a positive
constant C = Cq such that
1=2 ;
Mn q 6 Mn 1 + C Xk2
(A.1)
k6n
q
q 1=q
where Y q = (E|X | ) .
Let {(n ; n ∈ Nd } be a #eld of i.i.d. random variables independent of {Xn ; n ∈ Nd } such that P((n = ±1) = 12 .
De#ne S1; n = k6n; ((k) = 1 Xk ; S−1; n = k6n; ((k)=−1 Xk ; S̃ n = k6n ((k)Xk . Then S1; n + S−1; n = Sn ; S1; n −
S−1; n = S̃ n . Let M̃ n = maxk6n |S̃ n |. Then,
2M±1; n 6 max|S1; k + S−1; k | + max|S1; k − S−1; k | = Mn + M̃ n
k6n
k6n
and so
M1; n + M−1; n 6 Mn + M̃ n :
(A.2)
On the other hand, for a random variable Y = Y ((n ; Xn ; n ∈ Nd ), let EX Y denote the expectation of taking
over {Xn } for #xed {(n }, and let E( Y be de#ned similarly. Let Y X; q = (EX |Y |q )1=q . Noting that Mn contains
{Xn } only, we have,
Mn q = Mn X; q 6 M1; n X; q + M−1; n X; q
−
−
+
6 M1;+n X; q + M−1;
n X; q + M1; n X; q + M−1; n X; q ;
(A.3)
±
±
where M±1;
n = maxk6n S±1; n . Let
q−1
. = (M1;+n )q−1 =M1;+n X;
q
and
q−1
+
q−1
+
/ = (M−1;
=M−1;
n)
n X; q :
Then .X; p = /X; p = 1, where 1=p + 1=q = 1. Fix {(n }. Then the two sets {k: ((k) = 1} and {k: ((k) = −1}
+
+
are disjoint. It follows that M1;+n and M−1;
n are negatively associated, and then . and M−1; n are negatively
q−1
associated, since cx
is an increasing function of x on [0; ∞). So
+
+
+
+
EX .M−1;
n 6 [EX .][EX M−1; n ] 6 .X; p EX M−1; n = EX M−1; n :
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L.-X. Zhang, J. Wen / Statistics & Probability Letters 53 (2001) 259 – 267
Similarly, EX /M1;+n 6 EX M1;+n . It follows that
+
+
+
M1;+n X; q + M−1;
n X; q = EX .M1; n + EX /M−1; n
+
+
+
= EX (. − /)(M1;+n − M−1;
n ) + EX .M−1; n + EX /M1; n
+
+
+
6 . − /X; p M1;+n − M−1;
n X; q + EX M−1; n + EX M1; n
+
+
+
6 2M1;+n − M−1;
n X; q + EX (M−1; n + M1; n )
+
+
6 2M̃ n X; q + EX (M−1;
n + M1; n ):
(A.4)
−
−
−
M1;−n X; q + M−1;
n X; q 6 2M̃ n X; q + EX (M−1; n + M1; n ):
(A.5)
Similarly,
From (A.2) – (A.5), it follows that
Mn q 6 4M̃ n X; q + EX (M−1; n + M1; n )
6 4M̃ n X; q + EX M̃ n + EX Mn
6 5M̃ n X; q + E Mn = 5M̃ n X; q + Mn 1 :
That is,
q
(Mn q − Mn 1 )q 6 5q EX M̃ n :
Taking the expectation over {(n }, the previous inequality yields
q
(Mn q − Mn 1 )q 6 5q E M̃ n :
By noting that
q
E M̃ n = E E( max|S̃ k |q
k6n
q
6 2E[E( |S̃ n | ] 6 CE
q=2
Xk2
;
k6n
we complete the proof.
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