Acta Math. Hungar. 110 (4) (2006), 293–308.
A BERRY–ESSEEN THEOREM FOR WEAKLY NEGATIVELY DEPENDENT RANDOM VARIABLES AND ITS APPLICATIONS∗ J. F. WANG and L. X. ZHANG (Hangzhou)
Abstract. We provide uniform rates of convergence in the central limit theorem for linear negative quadrant dependent (LNQD) random variables. Let {Xn , P n = 1} be a LNQD sequence of random variables with EXn = 0, set Sn = n j=1 Xj and Bn2 = Var (Sn ). We show that
sup P x
2 n n3δ /(4+6δ) X Sn 2+δ < x − Φ(x) = O n−δ/(2+3δ) ∨ E|X | i Bn Bn2+δ i=1
under finite (2 + δ)th moment and a power decay rate of covariances. Moreover, by the truncation method, we obtain a Berry–Esseen type estimate for negatively associated (NA) random variables with only finite second moment. As applications, we obtain another convergence rate result in the central limit theorem and precise asymptotics in the law of the iterated logarithm for NA sequences, and also for LNQD sequences.
1. Introduction Two random variables X and Y are said to be negative quadrant dependent (NQD) [positive quadrant dependent (PQD)] if P (X > x, Y > y) − P (X > x)P (Y > y) 5 0 [= 0] for all x, y ∈ R, resp.¡ Note that two ¢ random variables X, Y are NQD [PQD] if and only if Cov f (X), g(Y ) 5 0 [= 0] for all real nondecreasing functions f and g, resp. A sequence {Xn , n = 1} is said to be linear negative [positive] quadrant dependent (LNQD) [(LPQD)] if for any disjoint finite subsets A, B P P ⊂ {1, 2, . . . } and any positive real numbers rj , i∈A ri Xi and j∈B rj Xj are NQD [PQD], resp. ∗ Research
supported by National Science Foundation of China (No. 10071072). Key words and phrases: Berry–Esseen theorem, linear negative quadrant dependence, negative association, convergence rate, central limit theorem, precise asymptotics, law of the iterated logarithm. 2000 Mathematics Subject Classification: 60F05, 60G50. c 2006 Akad´ 0236–5294/$ 20.00 ° emiai Kiad´ o, Budapest
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A finite family {X1 , . . . , Xn } of random variables is said to be negatively associated (NA) if any disjoint subsets A, B ⊂ {1, ¡ 2, . . . , n} and any A B nondecreasing functions f on R and g on R , Cov f (Xi , i ∈ A), g(Xj , ¢ j ∈ B) 5 0. A finite family {X1 , . . . , Xn } of random variables is said to be ¡ ¢ (positively) associated (PA) if Cov f (X1 , . . . , Xn ), g(X1 , . . . , Xn ) = 0 for any real coordinatewise nondecreasing functions f and g on Rn . An infinite family is NA [PA] if every finite subfamily is NA [PA], resp. The definition of NQD is given by Lehmann [7], the concept of LNQD is given by Newman [12], and the concept of NA is given by Joag-Dev and Proschan [6]. Because of their wide applications in multivariate statistical analysis and reliability theory, the notions of negative dependence have received more and more attention recently. For NA sequences, Newman [12] obtained the central limit theorem, Su et al. [16] obtained the functional central limit theorem, Pan [13] obtained a Berry–Esseen theorem, Shao and Su [15] obtained the law of iterated logarithm. It is easy to see that NA implies LNQD. Since LNQD is much weaker than NA, studying the limit theorems for LNQD sequences is of interest. Throughout this paper, let log x denote ln max(x, e), an = O(bn ) denote lim sup an /bn < ∞ and an = o(bn ) denote an /bn → 0. Let {Xn , n = 1} be a sequence of random variables with EXn = 0 and EXn2 < ∞, and put Sn = Pn 2 j=1 Xj and Bn = Var (Sn ). Zhang [18] has shown that LNQD sequences satisfy the central limit theorem, i.e. ¯ µ ¯ ¶ ¯ ¯ Sn ¯ (1.1) 4n := sup ¯P < x − Φ(x)¯¯ = o(1), Bn x Rx where Φ(x) = (2π)−1/2 −∞ exp(−t2 /2) dt denotes the standard normal distribution function, satisfying X © ª (i) u(n) := sup − Cov (Xj , Xk ) → 0 as n → ∞ and u(1) < ∞, k
=
j: |j−k| n
(ii) lim inf Bn2 /n > 0, n→∞
(iii) Bn−2
n X
¡ ¢ EXj2 I |Xj | = εBn → 0
for any ε > 0.
j=1
Up to now, there is no result on the convergence rate of 4n for LNQD sequences. Under finite third moment and the exponential decay rate of covariances, Birkel [1] and Pan [13] established a Berry–Esseen theorem for PA and NA sequences, respectively. However, it seems that the method of Acta Mathematica Hungarica 110, 2006
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Birkel [1] and Pan [13] cannot work for LNQD sequences. The main purpose of this paper is to establish Berry–Esseen type estimates for LNQD sequences. At first, we show (2.1) under finite (2 + δ)th moment and a power decay rate of covariances (see Theorem 2.1). We do not know whether this rate is optimal, but we note that our conditions are weaker than Pan’s [13] and Birkel’s [1] (see Remark 2.1). Moreover, by the truncation method, we obtain a Berry–Esseen type estimate for NA random variables with only finite second moment (see Theorem 2.3). As applications, we obtain another convergence rate result in the central limit theorem (see Theorem 4.2) and precise asymptotics in the law of the iterated logarithm for NA sequences, and also for LNQD sequences.
2. Main results Theorem 2.1. Suppose that {Xn , n = 1} is a LNQD sequence of random variables with EXn = 0, supEXn2 < ∞ and E|Xn |2+δ < ∞ for some n
δ ∈ (0, 1]. Assume that X © ª ¡ ¢ (I) sup − Cov (Xj , Xk ) = O n−λ k
=
for some
λ>1
j: |j−k| n
holds, then (2.1)
µ ¶ 2 n n3δ /(4+6δ) X 2+δ −δ/(2+3δ) ∆n = O n ∨ E|Xi | . Bn2+δ i=1
Corollary 2.1. Suppose that {Xn , n = 1} is a LNQD sequence of identically distributed random variables with EX1 = 0 and E|X1 |2+δ < ∞ for some δ ∈ (0, 1]. Assume that condition (I) and inf Bn2 /n > 0
(II)
n
hold, then δ
∆n = O(n− 2+3δ ).
(2.2)
Remark 2.1. Pan [13] [Birkel [1]] obtained the rate 4n = O(n−1/2 log n) ¡ ¢ [ = O n−1/2 (log n)2 ] for NA [PA] sequences fulfilling EXn = 0, X ¯ ¯ ¯ Cov (Xj , Xk )¯ = O(e−εn ) supE|Xn |3 < ∞, inf Bn2 /n > 0, sup n
n
k
=
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for some ε > 0, resp. Clearly, our conditions are weaker than Pan’s and Birkel’s. The following theorem is a useful tool in order to obtain a Berry–Esseen type estimate with only finite second moment. Theorem 2.2. Suppose that {Xn , n = 1} is a LNQD sequence of random variables with EXn = 0, supEXn2 < ∞ and E|Xn |2+δ < ∞ for some n
δ ∈ (0, 1]. Assume that condition (I) holds, then for any r > 0, Ã ! n rδ/2 X (log n) (2.3) ∆n = O (log n)−r/3 ∨ E|Xi |2+δ . Bn2+δ i=1 Now, by the truncation method, we can obtain a Berry–Esseen type estimate for NA random variables with only finite second moment. Set (n)
Yi
Sn∗ =
¢ ¡ = −Bn I(Xi < −Bn ) + Xi I |Xi | 5 Bn + Bn I(Xi > Bn ), Pn
(n) i=1 Yi
1 5 i 5 n,
¡ ¢ and (Bn∗ )2 = Var Sn∗ .
Theorem 2.3. Suppose that {Xn , n = 1} is a NA sequence of random variables with EXn = 0 and supEXn2 < ∞. Assume that condition (I) and n
Bn2 = (Bn∗ )2
(III)
hold, then for some δ ∈ (0, 1] and any r > 0 there exists a positive constant A such that ( n ¢ ¡ (log n)rδ/2 X 2 (2.4) ∆n 5 A EX I |X | > B i n i Bn2 i=1
) n ¢ (log n)rδ/2 X 2+δ ¡ −r/3 . + E|Xi | I |Xi | 5 Bn + (log n) Bn2+δ i=1
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3. Proof of main results In this and the next sections, C will denote a positive constant that may change from line to line. In order to prove the theorems, we need the following lemmas. Lemma 3.1 [9, Lemma 1]. Suppose that f1 (t) and f2 (t) are characteristic functions corresponding to the distribution function F1 (x) and F2 (x), respectively. Assume that F2 (x) has a bounded derivative on the real line. 1 Then for any T > 0, b > 2π , ¯Z T ¯ ¯ ¯ ¯ ¯ f (t) − f (t) 1 2 −itx (3.1) sup ¯ F1 (x) − F2 (x)¯ 5 b max sup ¯¯ hv (t)e dt¯¯ v=1,2 x it x −T Z ¯ ¯ ¯ F2 (x + y) − F2 (x)¯ dy, + bT sup x
5
|y| c(b)/T
where the constant c(b) depends only on b, µ µ ¶ ¶ 1 − |t| eita/T |t| < T 1 − |t| e−ita/T T T h1 (t) = , h2 (t) = 0 |t| = T 0
|t| < T |t| = T,
where the constant a depends only on b. Furthermore, we have (3.2)
¯Z ¯ sup ¯¯ x
T
−T
¯ ¯ ¯ ¯ f1 (t) − f2 (t) −itx ¯ hv (t)e dt¯ 5 sup ¯ F1 (x) − F2 (x)¯ it x
(v = 1, 2).
Lemma 3.2 [12, Theorem 10]. Suppose that X1 , X2 , . . . , Xn are LNQD or LPQD random variables with finite variance. Then for any real λ1 , . . . , λn ¯ ¯ ¶ Y µ X n n n ¯ ¯ X ¯ ¯ ¯ ¯ λk Xk − E exp (iλk Xk )¯ 5 |λk | |λj |¯ Cov (Xk , Xj )¯ . ¯E exp i ¯ ¯ k=1
k=1
k=1,j>k
Lemma 3.3 [14, Theorem 5.8]. Suppose that {Xn , n = 1} is a sequence of independent random variables with EXn = 0 and EXn2 < ∞. Then there exists a positive constant A such that ( ) n n X ¡ ¢ ¡ ¢ 1 X 1 3 ∆n 5 A EXi2 I |Xi | > Bn + 3 E|Xi | I |Xi | 5 Bn . Bn2 Bn i=1
i=1
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Lemma 3.4 [18, Lemma 3.3]. Suppose that {Xn , n = 1} is a LNQD sequence of random variables with EXn = 0. Then for any p > 1 there exists a positive constant D such that ¯X ¯p µX ¶p/2 n ¯ n ¯ 2 E ¯¯ Xk ¯¯ 5 DE Xk . k=1
k=1
Proof of Theorem 2.1. The method of the proof is based on Newman and Wright [11] and Wood [17] but the details are quite different. Let n = km, and ϕn (t) be the characteristic function of Sn /n1/2 . We define (k)
Wj
= (X(j−1)k+1 + · · · + Xjk )/k 1/2 ,
j = 1, . . . , m,
(k)
(k)
2 = Var W σkj ( j ), ψkj (t) is the characteristic function of Wj , j = 1, . . . , m and NA (x) is the normal distribution function with zero mean and variance ¯ ¯ √ ¯ (k) , 1 5 j 5 m} Bn2 /n. It easy to see that sup¯ N 0 (x)¯ = 1/ 2π. Let {W x
A
j
¯ (k) and W (k) be a sequence of independent random variables such that W j j have the same distribution for each j = 1, . . . , m. Then applying the triangle inequality to Lemma 3.1 (3.1) we get ¯ µ ¯ ¶ ¯ ¯ S n ¯ ∆n = sup ¯¯P < x − N (x) (3.3) A ¯ n1/2 x ¯ ¯Z ¯ ¯ T ϕn (t) − Qm ψkj (t/m1/2 ) ¯ ¯ j=1 hv (t)e−itx dt¯ 5 b max sup ¯ ¯ it v=1,2 x ¯ −T ¯ Z Qm Pm 2 2 1/2 ) − exp − ¯ T ( ¯ j=1 ψkj (t/m j=1 σkj t /(2m)) + b max sup ¯ it v=1,2 x ¯ −T ¯ ¯ −itx ¯ × hv (t)e dt¯ ¯ ¯Z ¯ T exp ( − Pm σ 2 t2 /(2m)) − exp ¡ − B 2 t2 /(2n)¢ n ¯ j=1 kj + b max sup ¯ it v=1,2 x ¯ −T ¯ ¯ bc2 (b) bc2 (b) ¯ × hv (t)e−itx dt¯ + √ := I1 + I2 + I3 + √ . ¯ 2πT 2πT Using Lemma 3.2 we get Z (3.4) I1 5 b
T
−T
Q ¯ 1/2 ) ¯¯ ¯ ϕn (t) − m j=1 ψkj (t/m ¯ ¯ dt ¯ ¯ t
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Z
299
½ 2n m o¾ Á X t (k) (k) − Cov (Wj , Wl ) t dt m
T
5b
−T l=1,j>l
bT 2 = 2m
(
m X
(k) Wj
Var (
) − Var
µX m
j=1
(k) Wj
¶) .
j=1
Using condition (I), for some λ > 1, we have µX ¶ m m X (k) (k) (3.5) Var (Wj ) − Var Wj j=1
2 = k
½X m X k−1 X k
j=1
Cov (X(j−1)k+i , X(j−1)k+l ) −
j=1 i=1 l=i+1 jk X
m−1 2 X = k
mk X ©
mk−1 X
mk X
i=1
l=i+1
¾ Cov (Xi , Xl )
ª − Cov (Xi , Xl )
j=1 i=(j−1)k+1 l=jk+1
=
m−1 k 2 X X k j=1
mk X ©
ª − Cov (Xt , Xjk+1−i )
i=1 t=jk+1
k mk X © ª 2(m − 1) X 5 − Cov (Xt , Xjk+1−i ) sup k j i=1
t=jk+1
∞
X
i=1
t: |t−s| i
2(m − 1) X 5 sup k s
©
=
ª − Cov (Xt , Xs )
∞
2C(m − 1) X −λ 5 i =O k i=1
µ
m−1 k
¶ .
By (3.4) and (3.5), we have µ (3.6)
I1 = O
T2 k
¶ .
Next, by Taylor expansion and (3.5) we get ¢ ¯¯ ¡ Pm 2 2 Z T ¯¯ 2 2 ¯ exp ( − j=1 σkj t /(2m)) − exp − Bn t /(2n) ¯ (3.7) I3 5 b ¯ ¯ dt ¯ t −T ¯ Acta Mathematica Hungarica 110, 2006
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¯ Pm 2 ¯¯ ¯ B2 ¯ n j=1 σkj ¯ 5 2b − ¯ ¯ t dt ¯ m 0 ¯ mk ( m µX ¶) µ 2¶ m bT 2 X T (k) (k) Var (Wj ) − Var Wj =O = . m k Z
T
j=1
j=1
Now, we estimate I2 . Using Lemma 3.1 (3.2) we get (3.8)
¯Z ¯ T E exp (i t Pm W ¯ (k) ) − exp ( − Pm σ 2 t2 /(2m)) 1/2 ¯ j=1 j=1 kj j m I2 = b max sup ¯ ¯ it v=1,2 x −T ¯ ¯ ¯ × hv (t)e−itx dt¯ ¯ ¯ Ã ¯ ! (k) ¯ ¯ S¯m ¯ ¯ 5 b sup ¯P < x − NB (x)¯ , 1/2 ¯ ¯ m x
P (k) ¯ (k) and NB (x) is the normal distribution function with where S¯m = m j=1 Wj P 2 zero mean and variance m j=1 σkj /m. ¯ (k) = EW (k) = 0, It is easy to see that E W j
j
k X 2 ¯ (k) )2 = E (W (k) )2 5 1 EX(j−1)k+i E (W 5 supEXn2 < ∞. j j k n i=1
So applying Lemma 3.3, for some δ ∈ (0, 1], we have ¯ Ã ¯ ! (k) ¯ ¯ ¯ S ¯ ¯ m sup ¯P < x − N (x) (3.9) ¯ B ¯ m1/2 x ¯ Ã µX ¶1/2 ! m m X A 2 (k) (k) 2 ¯ ) I |W ¯ |> E (W σkj 5 Pm 2 j j j=1 σkj j=1 j=1 + P m
(
m X
A
2 3/2 j=1 j=1 σkj
)
5
(
Pm
à ¯ (k) 3 I W j
E|
|
¯ (k) W j
|
|5
µX m j=1
m X
2A
2 j=1 σkj
Acta Mathematica Hungarica 110, 2006
)
(2+δ)/2
j=1
¯ (k) |2+δ . E |W j
2 σkj
¶1/2 !
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WEAKLY NEGATIVELY DEPENDENT RANDOM VARIABLES
Using Lemma 3.4 we get (3.10)
¯ (k) |2+δ = E |W (k) |2+δ = E |W j j
¯ k ¯2+δ ¯X ¯ ¯ E¯ X(j−1)k+i ¯¯ (2+δ)/2 k 1
i=1
¶(2+δ)/2 µ X k k 1X 1 2+δ 2 X(j−1)k+i 5C E |X(j−1)k+i | . 5 CE k k i=1
i=1
From (3.5) we get m X
(k)
Var (Wj
) − Var
j=1 m−1 2 X = k
µX m
¶ (k)
Wj
j=1 jk X
mk X ©
ª − Cov (Xi , Xl ) = 0.
j=1 i=(j−1)k+1 l=jk+1
Thus (3.11)
m X j=1
2 σkj =
m X j=1
(k)
Var (Wj
) = Var
µX m
¶ (k)
Wj
= Bn2 /k.
j=1
By (3.8), (3.9), (3.10) and (3.11), we have µ δ/2 X ¶ n k 2+δ (3.12) I2 = O E|X | . i Bn2+δ i=1 Next, by (3.3), (3.6), (3.7) and (3.12), we have ½ 2 ¾ n T 1 k δ/2 X 2+δ (3.13) ∆n 5 C + + 2+δ E|Xi | . k T Bn i=1 Choose T = k 1/3 . Then it follows that ¾ ½ n k δ/2 X (3.14) E|Xi |2+δ . ∆n 5 C k −1/3 + 2+δ Bn i=1 h 3δ i Finally, we choose k = n 2+3δ . Then it follows that ! Ã 2 n n3δ /(4+6δ) X 2+δ −δ/(2+3δ) E|Xi | . ∆n = O n ∨ Bn2+δ i=1
¤
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Remark 3.1. In the proof of Theorem 2.1, Lemma 3.4 is used to establish the moment bound ¯ ¯ 2+δ sup E ¯ Sn+k − Sk ¯ = O(n(2+δ)/2 ).
=
k 0
For LPQD sequences, we can use Theorem 1 in Birkel [2] which still holds for LPQD sequences (see Birkel [3]) to establish a similar moment bound, so an analogous approach could be used in order to establish a similar Berry– Esseen bound for LPQD random variables. Suppose that {Xn , n = 1} is a LPQD sequence of identically distributed random variables with EX1 = 0 0 and E|X1 |2+δ+δ < ∞ for some δ ∈ (0, 1] and some δ 0 > 0. Assume that (a) sup k
X
=
Cov (Xj , Xk ) = O(n−τ ) for τ = δ(2 + δ + δ 0 )/(2δ 0 ) > 1
j: |j−k| n
holds, then ∆n = O(n−δ/(2+3δ) ). £
Proof of Theorem 2.2. For any r > 0 and 0 < δ 5 1, we choose k = r¤ (log n) . (2.3) follows immediately from (3.14). ¤ (n)
(n)
Proof of Theorem 2.3. It easy to see that Y1 , . . . , Yn (n) Yi
(n) Yj
are bounded
NA random variables, and that Xi − and are nondecreasing functions corresponding to Xi and Xj , respectively. From the definition of NA random variables, we have (n)
Cov (Xi − Yi
(3.15)
(n)
, Yj
) 5 0,
i 6= j.
Similarly, we have (3.16)
(n)
Cov (Xj − Yj
, Xi ) 5 0,
i 6= j.
Thus, by (3.15) and (3.16), we have (3.17)
(n)
Cov (Xi , Xj ) 5 Cov (Yi
(n)
, Yj
),
i 6= j.
Because of EXj = 0, we have (3.18)
(n)
0 5 Var (Xj ) − Var (Yj
) = EXj2 I
¡
|Xj | > Bn
¢
¡ ¢ 2 ¡ ¢ + (EXj I |Xj | 5 Bn ) 5 2EXj2 I |Xj | > Bn . Acta Mathematica Hungarica 110, 2006
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Thus (3.19) Bn2
−
(Bn∗ )2
5
n ³ X
(n) Yj
Var (Xj ) − Var (
´
)
52
j=1
n X
¢ ¡ EXj2 I |Xj | > Bn .
j=1
If (Bn∗ )2 5 Bn2 /4, then Bn2 − (Bn∗ )2 = 3Bn2 /4. Thus, by (3.19) we get 15
n ¡ ¢ 8 X EXj2 I |Xj | > Bn , 2 3Bn j=1
which immediately yields (2.4). If (Bn∗ )2 > Bn2 /4, we have ¯ µ ¶ µ ∗ ¶¯ ¯ ¯ Sn Sn <x −P < x ¯¯ (3.20) ∆n 5 sup ¯¯P Bn Bn x ¯ µ ∗ ¶ µ ¶¯ ¯ Sn − ESn∗ xBn − ESn∗ xBn − ESn∗ ¯¯ ¯ + sup ¯P < −Φ ¯ Bn∗ Bn∗ Bn∗ x ¯ ¯ µ ¶ ¯ ¯ xBn − ESn∗ ¯ =: T1 + T2 + T3 . ¯ − Φ(x) + sup ¯Φ ¯ Bn∗ x First, (3.21)
T1 5
n n X ¢ ¡ ¢ ¡ 1 X P |Xj | > Bn 5 2 EXj2 I |Xj | > Bn . Bn j=1
j=1
Next, for any r > 0 and some δ ∈ (0, 1], applying Theorem 2.2 to 1 5 j 5 n} we get ( (3.22) (3.23)
−r/3
T2 5 C (log n)
(n)
E |Yj
+
n (log n)rδ/2 X
(Bn∗ )2+δ (n) 2+δ
− EYj
|
{Yj(n) ,
) E|
(n) Yj
−
(n) 2+δ EYj
|
,
j=1 (n) 2+δ
5 8E |Yj
|
¡ ¢ ¡ ¢ = 8{E|Xj |2+δ I |Xj | 5 Bn + Bn2+δ P |Xj | > Bn } ¡ ¢ ¡ ¢ 5 8E|Xj |2+δ I |Xj | 5 Bn + 8Bnδ EXj2 I |Xj | > Bn . Acta Mathematica Hungarica 110, 2006
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By combining (Bn∗ )2 > Bn2 /4, (3.22) and (3.23), we have ( n ¡ ¢ (log n)rδ/2 X 2 (3.24) T2 5 C EX I |X | > B j n j Bn2 j=1
) n ¡ ¢ (log n)rδ/2 X + E|Xj |2+δ I |Xj | 5 Bn + (log n)−r/3 . Bn2+δ j=1 Using Lemma 5.2 in Petrov [14] and condition (III), we have ¯ ¯ ¯ ESn∗ ¯ √ ∗ (3.25) T3 5 (Bn /Bn − 1)/ 2πe + √ . 2πBn∗ Because of EXj = 0, we have (3.26)
n n X ¯ ¯ ¢ ¢ ¡ ¡ 2 X ¯ ESn∗ ¯ 5 2 E|Xj |I |Xj | > Bn 5 EXj2 I |Xj | > Bn . Bn j=1
j=1
By (3.19) and (Bn∗ )2 > Bn2 /4, we have (3.27) ¡ ¢ Bn /Bn∗ − 1 = (Bn2 − (Bn∗ )2 )/( Bn + Bn∗ Bn∗ ) 5
n ¢ ¡ 8 X 2 . EX I |X | > B j n j 3Bn2 j=1
By combining (3.25), (3.26) and (3.27), we have (3.28)
T3 5
n ¢ ¡ C X EXj2 I |Xj | > Bn . 2 Bn j=1
Finally, by (3.20), (3.21), (3.24) and (3.28), we obtain (2.4).
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4. Applications The Berry–Esseen theorems established in Section 2 enable us to establish easily another convergence rate result in the central limit theorem (see Gut [5]) and precise asymptotics in the law of the iterated logarithm (see Gut and Spˇataru [4]) for NA sequences, and also for LNQD sequences. Theorem 4.1. Suppose that {Xn , n = 1} is a NA sequence of identically distributed random variables with EX1 = 0, and that conditions (I), (II) and ¡ ¢ 1+α+β (III) hold. Let α = −1 and EX12 log |X1 | < ∞ for some β > 0. Then ∞ X (log n)α
(4.1)
n
n=1
∆n < ∞.
Proof. From the definition of NA random variables, it follows easily that Bn2 = O(n), and from condition (II), it follows easily that n = O(Bn2 ). So, c2 n 5 Bn2 5 c1 n for 0 < c2 < c1 < ∞. For every fixed α = −1 and β > 0, we can choose r > 3(1 + α), 0 < δ 5 1 such that rδ/2 < β. Now, by Theorem 2.3 we get (4.2)
∞ X (log n)α
n
n=1
+C
∆n 5 C
∞ X (log n)α+rδ/2
n
n=1
∞ X (log n)α+rδ/2
n(2+δ)/2
n=1
+C
∞ X
EX12 I (|X1 | >
E|X1 |2+δ I (|X1 | 5
√ c2 n )
√ c1 n )
n−1 (log n)α−r/3 := K1 + K2 + C,
n=1
(4.3)
K1 5 C
∞ ∞ X (log n)α+β X
n
n=1
5C
∞ X
¡ ¢ EX12 I c2 j < X12 5 c2 (j + 1)
j=n
¡ ¢ (log j)1+α+β EX12 I c2 j < X12 5 c2 (j + 1)
j=1
¡ ¢ 1+α+β 5 CEX12 log |X1 | < ∞. Similarly, we have (4.4)
K2 5 C
∞ n X (log n)α+β X n=1
n(2+δ)/2
¡ ¢ E|X1 |2+δ I c1 (j − 1) < X12 5 c1 j
j=1 Acta Mathematica Hungarica 110, 2006
306
J. F. WANG and L. X. ZHANG
5C
∞ X
¡ ¢ j −δ/2 (log j)α+β E|X1 |2+δ I c1 (j − 1) < X12 5 c1 j
j=1
¡ ¢ α+β 5 CEX12 log |X1 | < ∞. By (4.2), (4.3) and (4.4), we immediately obtain (4.1).
¤
Theorem 4.2. Suppose that {Xn , n = 1} is a LNQD sequence of identically distributed random variables with EX1 = 0 and E|X1 |2+δ < ∞ for some δ ∈ (0, 1]. Assume that conditions (I) and (II) hold, then for α = −1, (4.1) holds. Proof. This follows immediately from Corollary 2.1. ¤ Now, we turn to establish the precise asymptotics in the law of the iterated logarithm for NA sequences and LNQD sequences. Theorem 4.3. Suppose that {Xn , n = 1} is a strictly stationary NA se¡ ¢ β+1 quence of random variables with EX1 = 0 and EX12 log |X1 | < ∞ for some β > 0. Let {an , n = 1} be a sequence of real numbers such that µ (4.5)
lim
n→∞
log log n n
¶1/2 an = γ ∈ (−∞, ∞)
and σ 2 := Var (X1 ) + 2
(4.6)
∞ X
Cov (X1 , Xj ) > 0.
j=2
Suppose that conditions (I), (II) and (III) hold, then (4.7)
lim √
p
ε2 − 2
ε& 2
∞ √ X p √ 1 P (|Sn | = ε Bn2 log log n + an ) = 2e− 2γ/σ . n
n=1
Proof. Let Ψ(x) = 1 − Φ(x) + Φ(−x). Applying the triangle inequality √ to (4.1) with α = 0 and x = ε log log n + an /Bn , we have (4.8)
∞ ´ X p 1 ¯¯ ³ ¯P |Sn | = Bn (ε log log n + an /Bn ) n
n=1
− Ψ(ε
p
¯ ¯ log log n + an /Bn )¯ < ∞.
Acta Mathematica Hungarica 110, 2006
WEAKLY NEGATIVELY DEPENDENT RANDOM VARIABLES
307
Under stationary assumption, it is easy to show that Bn2 → σ2, n → ∞ n (see Lin [10]). So, by condition (4.5), we have µ ¶ log log n 1/2 an √ = γ/σ. lim n→∞ n Bn / n
√ Thus, applying Lemma 3 (i) in Li et al. [8], let a0n = an /(Bn / n). We have ∞ p X p 1 2−2 lim (4.9) ε Ψ(ε log log n + an /Bn ) √ n ε& 2 n=1
= lim √
ε& 2
p
ε2 − 2
∞ X n=1
√ p √ √ 1 Ψ(ε log log n + a0n / n ) = 2e− 2γ/σ . n
Finally, by (4.8) and (4.9), we immediately obtain (4.7). ¤ Theorem 4.4. Suppose that {Xn , n = 1} is a strictly stationary LNQD sequence of random variables with EX1 = 0 and E|X1 |2+δ < ∞ for some δ > 0. Let {an , n = 1} be a sequence of real numbers such that (4.5) and (4.6) hold. Assume that conditions (I) and (II) hold, then (4.7) and ∞ X p 1 lim ε2 P (|Sn | = ε Bn2 log log n ) = 1 (4.10) ε&0 n log n n=1
hold. Proof. We only prove (4.10), the proof of (4.7) is similar to that of Theorem 4.3. Let Ψ(x) = 1 − Φ(x) + Φ(−x). Applying the triangle inequality to √ (4.1) with α = −1 and x = ε log log n we get ∞ ¯ X p p 1 ¯¯ ¯ (4.11) ¯P (|Sn | = εBn log log n ) − Ψ(ε log log n )¯ < ∞. n log n n=1
By Proposition 3.1 in Gut and Spˇataru [4], we have ∞ X p 1 2 (4.12) lim ε Ψ(ε log log n ) = 1. ε&0 n log n n=1
By (4.11) and (4.12), we have (4.10) ¤ Remark 4.1. Using analogous approach and Remark 3.1, it is easy to establish similar results for LPQD random variables. Acknowledgment. The authors are grateful to the anonymous referee for providing several helpful comments which led to a significant improvement of the article. Acta Mathematica Hungarica 110, 2006
308
J. F. WANG and L. X. ZHANG: WEAKLY NEGATIVELY DEPENDENT . . .
References [1] T. Birkel, On the convergence rate in the central limit theorem for associated processes, Ann. Probab., 16 (1988), 1685–1698. [2] T. Birkel, Moment bounds for associated sequences, Ann. Probab., 16 (1988), 1184– 1193. [3] T. Birkel, A functional certral limit theorem for positively dependent random variables, J. Multivariate Anal., 11 (1993), 314–320. [4] A. Gut and A. Spˇ ataru, Precise asymptotics in the law of the iterated logarithm, Ann. Probab., 28 (2000), 1870–1883. [5] A. Gut, Convergence rates in the central limit theorem for multidimensionally indexed random variables, Studia Sci. Math. Hungar., 37 (2001), 401–418. [6] K. Joag-Dev and F. Proschan, Negative association of random variables with applications, Ann. Statist., 11 (1983), 286–295. [7] E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist., 37 (1966), 1137– 1153. [8] D. Li, B. E. Nguyen and A. Rosalsky, A supplement to precise asymptotics in the law of the iterated logarithm, J. Math. Anal. Appl., 302 (2005), 84–96. [9] Z. Y. Lin, On generalizations of Berry–Esseen inequality for U -statistics, Acta Math. Applica. Sinica (in Chinese), 6 (1983), 468–475. [10] Z. Y. Lin, An invariance principle for negatively associated random variables, Chinese Sci. Bull., 42 (1997), 359–364. [11] C. M. Newman and A. L. Wright, An invariance principle for certain dependent sequence, Ann. Probab., 9 (1981), 671–675. [12] C. M. Newman, Asympotic independence and limit theorems for positively and negatively dependent random variables, in: Inequalities in Statistics and Probability (Tong, Y. L., ed., Institute of Mathematical Statistics, Hayward, CA, 1984), pp. 127–140. [13] J. M. Pan, On the convergence rate in the central limit theorem for negatively associated sequences, Chinese J. Appli. Probab. Statist., 13 (1997), 183–192. [14] V. V. Petrov, Limit Theorems of Probability Theory – Sequences of Independent Random Variables, Clarendon Press, Oxford (New York, 1995). [15] Q. M. Shao and C. Su, The law of iterated logarithm for negatively associated random variables, Stochastic Processes Appl., 83 (1999), 139–148. [16] C. Su, L. C. Zhao and Y. B. Wang, The moment inequalities and weak convergence for negatively associated sequences, Sci. China., 40A (1997), 172–182. [17] T. E. Wood, A Berry–Esseen theorem for associated random variables, Ann. Probab., 11 (1983), 1042–1047. [18] L. X. Zhang, A functional central limit theorem for asymptotically negatively dependent random fields, Acta Math. Hungar., 86 (2000), 237–259. (Received May 7, 2004; revised October 3, 2005) DEPARTMENT OF STATISTICS AND COMPUTING SCIENCE ZHEIANG GONGSHANG UNIVERSITY HANGZHOU 310035 P. R. CHINA E-MAIL:
[email protected]
Acta Mathematica Hungarica 110, 2006
DEPARTMENT OF MATHEMATICS ZHEJIANG UNIVERSITY HANGZHOU 310028 P. R. CHINA E-MAIL:
[email protected]
