Maximal inequalities and a law of the. iterated logarithm for negatively. associated random fields

Transcription

1 Maximal inequalities and a law of the arxiv:math/ v1 [math.pr] 17 Oct 2006 iterated logarithm for negatively associated random fields Li-Xin ZHANG Department of Mathematics, Zhejiang University, Hangzhou , China Abstract The exponential inequality ofthe maximum partial sums is a ey to establish the law of the iterated logarithm of negatively associated random variables. In the one-indexed random sequence case, such inequalities for negatively associated random variables are established by Shao (2000) by using his comparison theorem between negatively associated and independent random variables. In the multi-indexed random field case, the comparison theorem fails. The purpose of this paper is to establish the Kolmogorov exponential inequality as well a moment inequality of the maximum partial sums of a negatively associated random field via a different method. By using these inequalities, the sufficient and necessary condition for the law of the iterated logarithm of a negatively associated random field to hold is obtained. Key Words: negative association, law of the iterated logarithm, random field, Kolmogorov exponential inequality, the maximum partial sums. Abbreviated Title: LIL for NA Fields AMS 2000 subject classifications. Primary 60F15 Research supported by National Natural Science Foundation of China 1

2 1 Introduction and the law of iterated logarithm A finite family of random variables {X i ;1 i n} is said to be negatively associated if for every pair of disjoint subsets A and B of {1,2,,n}, Cov{f(X i ;i A),g(X j ;j B)} 0 (1.1) whenever f and g are coordinatewise non-decreasing and the covariance exists. An infinite family is negatively associated if every finite subfamily is negatively associated. The concept of the negative association was introduced by Alam and Saxena (1981) and Joag-Dev and Proschan (1983). As pointed out and proved by Joag-Dev and Proschan (1983), a number of well-nown multivariate distributions possess the NA property. Many investigators discuss the properties and limit theorems of negatively associated random variables. We refer to Joag-Dev and Proschan (1983) for fundamental properties, Newman (1984) for the central limit theorem, Matula (1992) for the three series theorem, Su, et al. (1997) for the moment inequality, Roussas (1996) for the Hoeffding inequality, and Shao (2000) for the Rosenthaltype maximal inequality and the Kolmogorov exponential inequality. Shao and Su (1999) established the law of the iterated logarithm for negatively associated random variables with finite variances. Theorem A Let {X i ;i 1} be a strictly stationary negatively associated sequence with EX 1 = 0, EX 2 1 < and σ2 := EX i=2 E(X 1X i ) > 0. Let S n = n i=1 X i. Then Here and in the squeal of this paper, logx = ln(x e) S n = σ a.s. (1.2) (2nloglogn) 1/2 Let {X n ;n N d } be a field of random variables, where d 2 is a positive integer, N d denotes the d-dimensional lattice of positive integers. Through this paper, for n = (n 1,,n d ) N d, = ( 1,, d ) N d and m N, we denote n = n 1 n d, n = n n d, n = ( 1 n 1,, d n d ) and m = ( 1 m,, d m). Also, n (resp. n) means i n i (resp. i n i ), i = 1,2,,d. Denote by S n = n X and 1 = (1,,1) N d. It is nown that, if {X n } is a field of i.i.d.r.v.s, then S n (2d n loglog n ) 1/2 = (EX2 1 )1/2 a.s. (1.3) 2

3 if and only if EX 1 = 0 and EX1 2logd 1 ( X 1 )/loglog( X 1 ) <, where n means n 1,,n d. When {X n ;n N d } is a negatively associated field of random variables, Zhang and Wen (2001a) and Zhang and Wang (1999) established the wea convergence, the law of large numbers and the complete convergence similar to those for fields of independent random variables. This paper is to establish the law of the iterated logarithm similar to (1.3) for a negatively associated random field. Theorem 1.1 Let d 2 be a positive integer, and {X n ;n N d } be a wealy stationary negatively associated field of identically distributed random variables satisfying EX 1 = 0 and EX 2 1log d 1 ( X 1 )/loglog( X 1 ) <. (1.4) Denote by Υ(j i) = Cov(X j,x i ) and σ 2 = j Zd Υ(j). Then S n = σ a.s. (1.5) (2d n loglog n ) 1/2 Theorem 1.2 Let d 2 be a positive integer, and {X n ;n N d } be a negatively associated field of identically distributed random variables satisfying (1.4). Then S n (2d n loglog n ) 1/2 (EX2 1) 1/2 a.s. The following theorem tells us that the condition (1.4) is necessary for the law of the iterated logarithm to hold. Theorem 1.3 Let d 1 be a positive integer, and {X n ;n N d } be a negatively associated field of identically distributed random variables. If then (1.4) holds. ( ) S n P (2d n loglog n ) 1/2 < > 0, (1.6) In showing the law of the iterated logarithm, a main step is to establish the exponential inequalities. In the case of d = 1, such exponential inequalities for negatively associated random variables are established by Shao (2000) by using a comparison theorem between negatively associated random variables and independent random variables. However, if d 2, such a comparison theorem fails for the maximum partial sums (cf., Bulinsi and 3

4 Suquet 2001). In section 2, we establish a Kolmogorov type exponential inequality as well as a moment inequality of the maximum partial sums of a negatively associated field via a different method. Theorems are proved in Section 3. 2 Moment inequalities and exponential inequalities First, we have the following moment inequalities and exponential inequalities for the partial sums. Lemma 2.1 Let p 2 and let {Y ; n} be a negatively associated field of random variables with EY = 0 and E Y p <. Then E {( Y i p 2(15p/lnp) p i n i n EY 2 i ) p/2 + E Y i p}. Lemma 2.2 Let {Y ; n} be a negatively associated field of random variables with zero means and finite second moments. Let T = i Y i and Bn 2 = n EY 2. Then for all x > 0 and a > 0, P(T n x) P ( max Y > a ) ( x 2 ) +exp n 2(ax+Bn) 2. (2.1) Lemma 2.3 Let {Y ; n} be a negatively associated field of random variables with EY = 0 and E Y 3 <. Denote by T = i Y i and Bn 2 = n EY 2. Then for all x > 0, P(T n xb n ) ( 1 Φ(x+1) ) 6B 2 n 12B 3 n i n 1 i j n E(Y i Y j ) E Y 3 (2.2) n where Φ(x) is the distribution of a standard normal variable. Proofs of Lemmas : In the case of d = 1, Lemma 2.1 is proved by Shao (2000), and Lemma 2.3 is proved by Shao and Su (1999). Also, (2.1) follows from the following inequality easily: P(T n x) P ( max Y > a ) { x ( x +exp n a a + B2 ) ( n xa a 2 ln +1) } Bn 2. The later is proved by Su, et al.(1997). Since Lemmas do not involve the partial order of the index set, so them are valid for d 2 also. In fact, when d 2, there is a 4

5 one-one map π : { : n} {1,2,, n }. By noting that {Y π 1 (i) : i = 1,, n } is a negatively associated sequence and n i=1 ( ) = n ( ), max ny = max i n Y π 1 (i), the results follow. In a same may, one can extend (1.2) of shao (2000) to the case of d 2. Lemma 2.4 Let {Y ; n} be a negatively associated field and {Y ; n} be a field of independent random variables such that for each, Y and Y have the same distribution. Then Ef( Y ) Ef( n ny ) for any convex function f on R, whenever the expectations exist. It shall be mentioned that it is impossible to find a one-one map π : { : n} {1,2,, n } such that m i=1 Y π 1 (i) = m Y for all m n. So, the inequalities for maximum partial sums can not be extended directly. For maximum partial sums, Zhang and Wen (2001a) established two moment inequalities. Proposition 2.1 (Zhang and Wen 2001a) Let p 2, and let {Y ; n} be a negatively associated field of random variables with EY = 0 and E Y p <. Suppose that {ǫ ; n} is a field of i.i.d.r.v.s with P(ǫ = ±1) = 1/2. Also, suppose that {ǫ ; n} is independent of {Y ; n}. Denote by T = i Y i, M n = max n T, T = i Y i, Mn = max n T and X p = (E X p ) 1/p. Then M n p 5 M n p + M n 1, (2.3) and there exists a constant A p,d depending on p and d such that E M n p A p,d { (E M n ) p + ( n EY 2 ) p/2 + E Y p}. (2.4) n Proposition 2.2 (Zhang and Wen 2001a) Let {Y ; n} be a strictly stationary negatively associated field of random variables with EY 1 = 0 and 0 < EY 2 1 <. Denote by T = i Y i. Then there exists a constant K, depending only on d, such that n 1 Emax n T2 KEY 1. 2 (2.5) 5

6 The next theorem gives the estimate of EM n in (2.4) for a non-stationary negatively associated field. Theorem 2.1 Let {Y ; n} be a negatively associated field of random variables with EY = 0 and EY 2 <. Denote by T = i Y i, M n = max n T. Then there is a constant C d depending only on d such that EM n C d ney 2. (2.6) Proof. We will prove(2.6) byinductionon d. We usetheargument of Utev andpeligrad (2003). For each n, define a n = sup Y Emax m n m Y / n EY 2 1/2 (2.7) where the supremum is taen over all fields Y := {Y i } of square integrable centered negatively associated random variables. Fix such a random field {Y i } and in addition without loss of generality assume that EY 2 = 1. n Let M be a positive integer that will be specified later. Let f(x) = (( M 1/2 ) x) M 1/2. For n define: ξ = f(y ) Ef(Y ) and η = Y ξ. Then both {ξ n } and {η n } are negatively associated fields. Since ne η 2M 1/2 n EY 2 = 2M1/2, we get Emax m n Y m Emax m n ξ m +2M1/2. To estimate Emax m ξ, we shall use a blocing procedure. m n Write i = (i 1,,i d 1 ). Tae m 0 = 0 and define the integers m recursively by m m = min m : m > m 1, Eξ 2 i,j > 1 M. j=m 1 +1i n 6

7 Note that, if we denote by l the number of integers produced by this procedure, i.e.: m 0, m 1,..., m l 1, we have 1 l 1 m =1j=m 1 +1 i,j Write S m, = m j=m 1 +1 that is S m,l = n d j=m l 1 +1 Emax m n ξ m E max max 1 lm n =: I +II. j=1 Eξ 2 i,j > l 1 M so that l M. i m ξ i,j for 1 l 1 and for convenience, m l = n d i m ξ i,j S m,j. It is obvious that +E max max 1 lm n max j ξ m 1 <j<m,t t=m 1 +1 m We evaluate the two terms in the right hand side of (2.8) separately. By the induction hypothesis and Cauchy-Schwartz inequality l l I Emax S m, C d 1 E ( m ) 2 ξ m n i,j =1 l C d 1 =1 m j=m 1 +1 =1 i n j=m 1 +1 i neξ 2 i,j C d 1M 1/2 i n Eξ 2 i K d 1M 1/2. (2.8) When d = 1, the above inequality obviously holds with C 0 = 1, since max m n does not appear. To estimate II we notice that By (2.4), we obtain By the definition of m, (II) 4 Emax l Emax =1 max j ξ m nm 1 <j<m,t t=m 1 +1 m max ξ m nm 1 <j<m,t t=m 1 +1 m A 4,d E4 max m n + m 1 t=m 1 +1 n j max j Eξ 4,t + 4 ξ m 1 <j<m,t t=m 1 +1 m m 1 t=m 1 +1 n m 1 t=m 1 +1 n Eξ 2,t 1 M, 7 4 Eξ 2,t. 2.

8 so that by using the notation (2.7) for a n and the definition of ξ we obtain m 1 t=m 1 +1 n Eξ 2,t 4 M m 1 t=m 1 +1 n Eξ 4,t 4 M 2 and E 4 max Hence we obtain max j ξ m nm 1 <j<m,t t=m 1 +1 m a4 n m 1 t=m 1 +1 n { a (II) 4 4 A n 4,d M + 4 M + 1 }. M Eξ 2,t 2 a4 n M 2. Now by (2.8) and the estimates for I and II we get Emax m n Y (A 4,d/M) 1/4 a n +(5A 4,d /M) 1/4 +C d 1 M 1/2 +2M 1/2. m Therefore, by the definition of a n, a n (A 4,d /M) 1/4 a n +(5A 4,d /M) 1/4 +C d 1 M 1/2 +2M 1/2. Letting M = [16A 4,d ]+1 yields a n 2+2(C d 1 +2)M 1/2. The proof is now completed. Combining (2.4) and (2.6) we obtain the following result on the moment inequality for the maximum partial sums. Theorem 2.2 Let p 2, and let {Y ; n} be a negatively associated field of random variables with EY = 0 and E Y p <. Then there exists a constant C p,d depending on p and d such that Emax { Y p ( Cp,d m n m n EY 2 ) p/2 + E Y p}. (2.9) n In the case of d = 1, inequality (2.9) is first obtained by Shao (2000). Before that, Su et al (1997) proved that m { Emax Y p (nmax Cp EY 2 ) p/2 +nsup E Y p}. m n n =1 8 n

9 With the same proof, one can show that (2.9) of a wealy dependent random field with lim ρ n < 1 (for definition, see Peligrad and Gut (1999)) and a ρ -mixing random field (for definition, see Zhang and Wang (1999) or Zhang and Wen (2001a)). Now, we begin to establish to following Kolmogorov type exponential inequality for maximum partial sums. Theorem 2.3 Let {Y ; n} be a negatively associated field of random variables with EY = 0 and Y b a.s for some 0 < b <. Denote by T = i Y i, M n = max n T and Bn 2 = n EY 2. Then for all x > 0, P(M n 2EM n 20x) 2 d+1 exp ( x 2 ). (2.10) 2(bx+Bn) 2 Proof. Let {ǫ ; n} is a field of i.i.d.r.v.s with P(ǫ = ±1) = 1/2. Also, assume that {ǫ ; n} is independent of {Y ; n}. Denote by T n = n ǫ Y, T n,1 = n,ǫ =1 Y and T n,2 = n,ǫ = 1 Y. First, we show that Ee Mn 2 d e 2 Mn 1 Ee 10 T n 2 d+1 e 2 Mn 1 By the Lévy inequality, we have Ee 20ǫ Y. (2.11) P{ M n x} = E Y P ǫ { M n x} 2 d E Y P ǫ { T n x} 2 d P{ T n x}, x 0, n where E Y ( ) = E[ ǫ, N d ], and E ǫ, P ǫ etc are defined similarly. Then So, from (2.3) it follows that Ee Mn = 1+EM n + 1+ M n M n 1 + = 1+ M n q=2 Ee 10 M n 2 d Ee 10 T n. EM q n q! (5 M n q + M n 1 ) q q=2 q! ( ) 2 5 Mn + M n q q 1 q=2 q! E(10 M n +2 M n 1 ) q q=2 q! E(10 M n +2 M n 1 ) q q=1 q! { = E 1+ = Ee 10 M n+2 M n 1 2 d e 2 Mn 1 Ee 10 T n. 9 (10 M n +2 M n 1 ) q } q=1 q!

10 Note that Ee 10 T n = Ee 10T n,1 10T n,2 1 2 E( e 20T n,1 +e 20T n,2 ). For fixed {ǫ ; n}, we have by Lemma 2.4 or the definition (1.1), E Y e 20T n,1 E Y e 20Y = E Y e 20ǫ Y ǫ =1 E Y e 20ǫ Y ǫ =1 ǫ =1 ǫ = 1E Y e 20ǫY = n E Y e 20ǫ Y, since E Y e 20ǫ Y e E Y (20ǫ Y ) = 1. It follows that Ee 20T n,1 = E ǫ E Y e 20T ( n,1 E ǫ E Y e 20ǫ ) Y n = E ǫ E Y e 20ǫ Y = n nee 20ǫ Y. Similarly, Ee 20T n,2 n Ee 20ǫ Y = nee 20ǫ Y. It follows that Ee 10 T n nee 20ǫ Y. Similarly, Ee 10 T n nee 20ǫ Y. (2.11) is proved. Now, from (2.11) it follows that for any x > 0 and t > 0, P(M n 2EM n 20x) e 20tx 2t Mn 1 Ee tmn 2 d+1 e 20tx nee 20tǫ Y. Since Ee 20tǫ Y { = E 1+20tǫ Y + e20tǫ Y 1 20tǫ Y (ǫ Y ) 2 (ǫ Y ) 2} 1+(e 20tb 1 20tb)b 2 EY 2 exp { (e 20tb 1 20tb)b 2 EY 2 }, it follows that P(M n 2EM n 20x) 2 d+1 e 20tx exp { (e 20tb 1 20tb)b 2 B 2 n}. 10

11 Letting 20t = 1 xb b log(1+ ) yields Bn 2 { x P(M n 2EM n 20x) 2 d+1 exp b (x b + B2 n { 2 d+1 exp x 2 } 2(bx+Bn 2). xb )log(1+ b2 Bn 2 } ) 3 Proofs of the laws of iterated logarithm We need two more lemmas. Lemma 3.1 Let d 2 and let {Y ; N d } be a negatively associated field of identically distributed random variables with EY 1 = 0 and EY 2 1 log d 1 ( Y 1 )/loglog( Y 1 ) <. Denote by T n = n Y and M n = max n T. Then M n (2d n loglog n ) 1/2 20(EY 2 1 )1/2 a.s. (3.1) Proof. Let0 < ǫ < 1beanarbitrarybutfixednumber. Letb m = ǫ 40 (EY 2 1 )1/2 (m/loglogm) 1/2, f (x) = ( b ) (x b ), g (x) = x f (x). Define First, we show that Y = f (Y ) Ef (Y ), Ŷ = g (Y ), T n = ny, M n = max T, Tn = Ŷ. n n T n E T n 0 a.s. as n. (3.2) (2 n loglog n ) 1/2 Since n E Ŷ (2 n loglog n ) 1/2 n E Y I{ Y b } (2 n loglog n ) 1/2 n C (loglog ) 1/2 EY1 2I{ Y 1 b } (2 n loglog n ) 1/2 n C 1/2 EY1 2I{ Y 1 b } n 1/2 = o(1) n 1/2 n 1/2 0 as n, (3.3) 11

12 it is enough to show that n Ŷ 0 a.s. (3.4) (2 n loglog n ) 1/2 For n = (n 1,,n d ) N d, let I(n) = { = ( 1,, d ) : 2 n i 1 i 2 n i 1,i = 1, d}. (3.4) will be true if we have and I(n)Ŷ+ (2 n loglog2 n 0 a.s. (3.5) ) 1/2 I(n)Ŷ (2 n loglog2 n 0 a.s. (3.6) ) 1/2 We show (3.5) only since (3.6) can be showed similarly. Let α m := α(m) = (2mloglogm) 1/2 and Z = (Ŷ + ) α. It is easily seen that Z = 0 if Y b, Z = Y b if b Y b +α and Z = α if Y b +α. Also, {Z ; N d } is a negatively associated field of random variables. Obviously, P(Ŷ n + Z n) P(Y n α n ) C n n (logm) d 1 P(Y 1 α m ) m=1 CEY 2 1 logd 1 ( Y 1 )/loglog( Y 1 ) <. Also by (3.3), using the notation 2 n = (2 n 1,,2 n d) for n = (n 1,,n d ) N d, one has that, I(n) EZ (2 n loglog2 n ) 1/2 2 n E Ŷ (2 n loglog2 n 0. ) 1/2 So, (3.5) is equivalent to Let I(n) (Z EZ ) (2 n loglog2 n 0 a.s. (3.7) ) 1/2 Λ(n) = I(n) EY1 2I{b < Y 1 2α }. α 2 12

13 Note that b < α if is large enough. From Lemma 2.1, it follows that for n large enough and any δ > 0, P ( I(n) (Z EZ ) δ(2 n loglog2 n ) 1/2) C E I(n) (Z EZ ) 4 (2 n loglog2 n ) 2 C(α(2 n )) 4{ ( E Z 2) 2 + I(n) C(α(2 n )) 4{ ( + I(n) I(n) +C(α(2 n )) 4{ ( + I(n) C {Λ 2 (n)+ { ( +C I(n) I(n) E Z 4} EY 2 1I{b < Y 1 α +b } ) 2 } EY1I{b 4 < Y 1 α +b } I(n) α 2 P( Y 1 α +b ) ) 2 } α 4 P( Y 1 α +b ) I(n) EY1 4I{b < Y 1 2α }} α 4 P( Y 1 α ) ) 2 + I(n) } P( Y 1 α ). Obviously, { ( n C n I(n) P( Y 1 α ) ) 2 + I(n) } P( Y 1 α ) P( Y 1 α n ) CEY 2 1 logd 1 ( Y 1 )/loglog( Y 1 ) <. Also, EY 4 1 I{b < Y 1 2α } EY 4 1 I{b n < Y 1 2α n } C C C n I(n) α 4 (logm) d 1EY 1 4I{ Y 1 2α m } m=1 m (logm) d 1 m=1 =1m= =1 =1 α 4 m n EY 4 1 I{2α 1 < Y 1 2α } α 4 m (logm) d 1 (mloglogm) 2EY 4 1 I{2α 1 < Y 1 2α } α 4 n (log) d 1 (loglog) 2(loglog)EY 2 1 I{2α 1 < Y 1 2α } CEY 2 1 log d 1 ( Y 1 )/loglog( Y 1 ) <, 13

14 and, similarly to (3.8) of Li and Wu (1989) we have Λ 2 (n) <. It follows that for arbitrary δ > 0, P ( (Z EZ ) δ(2 n loglog2 n ) 1/2) <, n I(n) n which implies (3.7) by the Borel-Cantelli lemma. Thus (3.2) holds. Now, by applying (2.10) to x = (1 + 2ǫ)(EY1 2)1/2 (2d n loglog n ) 1/2 and b = 2b n, it follows that P ( M n 2EM n 20(1+2ǫ)(EY 2 1) 1/2 (2d n loglog n ) 1/2) 2 d+1 exp{ (1+ǫ)dloglog n } 2 d+1 (log n ) (1+ǫ)d. For θ > 1 and m N d, let N m = ([θ m 1 ],,[θ m d]). It follows that P ( M Nm 2EM Nm 20(1+2ǫ)(EY1) 2 1/2 (2d N m loglog N m ) 1/2) m C m (1+ǫ)d C i d 1 i (1+ǫ)d <. m i=1 Notice EM Nm C N m 1/2 by Theorem 2.1. From the Borel-Cantelli lemma, it follows that m M Nm (2d N m loglog N m ) 1/2 20(1+2ǫ)(EY 2 1) 1/2 +2limsup m = 20(1+2ǫ)(EY 2 1) 1/2. EM Nm (2d N m loglog N m ) 1/2 So, we have limsup M n (2d n loglog n ) 1/2 max N m 1 n N m M Nm (2d N m 1 loglog N m 1 ) 1/2 20θ d/2 (1+2ǫ)(EY 2 1 )1/2 a.s. (3.8) Finally, from (3.2) and (3.8) it follows that (3.1) holds. 14

15 Lemma 3.2 Let {X ; N d } be a negatively associated field of bounded random variables with EX = 0 for all N d. Denote by S n = n X. Then for any δ > 0, max δn S n+ S n (2d n loglog n ) 1/2 80d(δ(1+2δ) d ) 1/2 sup(ex 2 )1/2 (3.9) Proof. Assume that X b a.s. with 0 < b <. Denote β = sup (EX 2 )1/2. From Theorem 2.2 it follows that Emax ( X ) 2 C n β 2. (3.10) m n m For fixed n and m n, let Y = 0 for m and Y = X otherwise, and let T = i Y i for m+δn, Bm+[δn] 2 = m+[δn] EY 2. Then for m n, B 2 m+[δn] ( m+[δn] m ) β 2 δd(1+δ) d n β 2. So by Theorem 2.3, Note that by (3.10) we have P ( max δn S m+ S m 2Emax δn S m+ S m 20x ) = P ( max T E max T 20x ) m+[δn] m+[δn] { 2 d+1 x 2 } exp 2(bx+Bm+[δn] 2 ) { 2 d+1 exp x 2 2(bx+δd(1+δ) d n β 2 ) }. (3.11) max Emax S m+ S m 2E max S 2 ( ) 1/2 E max m n δn n+[δn] n+[δn] S2 K ( n+[δn] β 2) 1/2 K(1+δ) d/2 n 1/2 β. From (3.11) it follows that for n large enough ( max P max S m+ S m 20d(1+2ǫ) ( 2δ(1+δ) d d 2 β 2 n loglog n ) ) 1/2 m n δn 2 d+1 exp { (1+ǫ)dloglog n }. 15

16 Now, let I p = {p+ : [δn]}. Then there are at most [(δ 1 +1) d ]+1 such I p s whose union covers { : n}. It follows that ( P max max S m+ S m (1+2ǫ)80d ( δ(1+2δ) d β 2 n loglog n ) ) 1/2 m n δn ( ( 1 ) ( δ +1)d +1 max P max max S m+ S m p m I p δn 40d(1+2ǫ) ( 4δ(1+2δ) d β 2 n loglog n ) ) 1/2 4( 1 ( δ +1)d max P max S m+ S m m n 2δn 20d(1+2ǫ) ( 4δ(1+2δ) d β 2 n loglog n ) 1/2 ) 4( 1 δ +1)d 2 d+1 exp ( (1+ǫ)dloglog n ). If we choose n p = ([θ p 1 ],,[θ p d]), then the sum of the above probability is finite. And then by the Borel-Cantelli lemma, p max m np max δnp S m+ S m (2d n p loglog n p ) 1/2 80d ( δ(1+2δ) d) 1/2 β a.s., which implies (3.9) easily. Now, we turn to the Proof of Theorem 1.1: We can assume σ > 0, for otherwise, we can consider the field {X n +ǫz n ;n N d } instead, where {Z n ;n N d } is a field of i.i.d. standard normal random variables and ǫ > 0 is an arbitrary number. First we show that S n σ a.s. (3.12) (2d n loglog n ) 1/2 For b > 0, let g b (x) = ( b) (x b) and h b (x) = x g b (x). Then g b (x) and h b (x) are both non-decreasing functions of x. Let X = g b (X ) Eg b (X ), X = h b (X ) Eh b (X ), S n = n X andm n = max n S. AnddefineŜ, M n similarly. ThenX = X + X and, {X ; N d } and { X ; N d } are both negatively associated fields of identically distributed random variables with EX = E X = 0 and X 2b. Then by Lemma 3.1, Ŝn (2d n loglog n ) 1/2 20(E X 2 1 )1/2 20 ( EX 2 1 I{ X 1 b} ) 1/2 0 a.s. as b. (3.13) So it suffices to show that for any ǫ > 0, S n (2d n loglog n ) 1/2 (1+ǫ)2 σ a.s. (3.14) 16

17 if b is large enogh. Let I m = { : 1 i m,i = 1,,d} and Y = i I m X ( 1)m+i. Since E( i I m X ( 1)m+i ) 2 = ES 2 1m /md σ 2 and E (Y ) 2/m X d ( 1)m+i 2EX1I{ X 2 1 b} 0 as b, i I m we can choose b and m large enough such that supey 2 (1+ǫ)2 m d σ 2. For θ > 1, let N = [θ ] =: ([θ 1 ],,[θ d]). From (2.1), it follows that P( S N m x) = P( i N Y i x) 2exp Let x = (1+ǫ) 2 σ(2dm d N loglog N ) 1/2. It follows that { x 2 } 2(2m d bx+ i N EYi 2). P ( S N m (1+ǫ) 2 σ(2dm d N loglog N ) 1/2) C C P ( S N m (1+ǫ)(2d EYi 2 loglog N ) 1/2) i N exp { (1+ǫ)dloglog N } (1+ǫ)d C i d 1 i (1+ǫ)d <. i=1 From the Borel-Cantelli lemma, it follows that = limsup So, from Lemma 3.2 it follows that limsup limsup +limsup S N m (2d N m loglog N m ) 1/2 S N m (2dm d N loglog N ) 1/2 (1+ǫ)2 σ a.s. S n (2d n loglog n ) 1/2 max N m n N +1 m S N m (2d N m loglog N m ) 1/2 max N m n N +1 m S n (2d N m loglog N m ) 1/2 S n S N m (2d N m loglog N m ) 1/2 (1+ǫ)σ +80d ( (θ 1)(1+2(θ 1)) d) 1/2 (EX 2 ) 1/2 17 a.s.

18 Letting θ 1 completes the proof of (3.14). Next we show that S n σ a.s. (2d n loglog n ) 1/2 Due to (3.13), it suffices to show that for any 0 < ǫ < 1/9, for b large enough. S n (1 9ǫ)σ a.s. (3.15) (2d n loglog n ) 1/2 For N, let m = [2 1+ǫ ], p = [ ǫ ], N = (m + p ) 4. For N d and an integer q, let m = (m 1,,m d ), p = (p 1,,p d ), N = (N 1,,N d ) and q = ( q 1,,q d ). We first show the following equality P ( S N (1 7ǫ)σ(2d N loglog N ) 1/2) =. (3.16) Set I i = I i () = {(i 1)(m +p )+1 m (i 1)(m +p )+m }, A = 1 i 4I i and B = {m : m N }\A. Then CardA = 4 m N and CardB = N CardA = o( N ). Let v i,1 = j I i X j and Clearly, S,1 = v i,1 = X j, S,2 = X j. 1 i 4 j A j B and S N = S,1 +S,2 j B EX 2 j/ N EX 2 1CardB / N 0,. From (2.1), it follows that for any ǫ > 0 P ( S,2 ǫσ(2d N loglog N ) 1/2) 2 C { exp ǫ 2 σ 2 2d N loglog N 2{bǫσ(2d N loglog N ) 1/2 + j B EX 2 j } exp { 3dloglog N } <. } 18

19 Thus in order to prove (3.16) is enough to show that P ( S,1 (1 6ǫ)σ(2d N loglog N ) 1/2) =. (3.17) Let B 2 = 1 i 4Ev2 i,1. Since and 1 i 4 E 1 i 4 E ( j I i X j ( vi,1 j I i X j ) 2 = 4 ES 2 m 4 m σ 2 N σ 2 ) 2/ N CEX 2 I{ X b} 0 as b, for b and large enough B 2 (1 ǫ)2 N σ 2. From Lemma 2.3, it follows that P ( S,1 (1 6ǫ)σ(2d N loglog N ) 1/2) P ( S,1 (1 5ǫ)(2dB 2 N loglog N ) 1/2) { 1 Φ ( 1+(1 5ǫ)(2dloglog N ) 1/2)} J,1 J,2 where Obviously, J,1 = O(1) N 1 E(v i,1 v j,1 ), 1 i j 4 J,2 = O(1) N 3/2 E v i, i 4 { ( 1 Φ 1+(1 5ǫ)(2dloglog N ) 1/2)} C (log N ) (1 4ǫ)d C (1 2ǫ)d =, and by Lemma 2.1, Also, J,2 C p N 3/2 4 ( m 3/2 (2b) 3 + m (2b) 3 ) C J,1 = O(1) N 1 2 <. Cov(X i,x j ) 1 i j N,j i p = O(1) N 1 Cov(X i,x j ) 1 i j N,j i p = O(1) Cov(X 1,X j+1 ) = O(1) Cov(X 1,X j+1 ), p j N N 1 <j N 19

20 since p i = o(n i ) for i = 1,,d. It follows that J,1 = O(1) j Cov(X 1,X j+1 ) <. Hence (3.17) is proved. Now, let C = {m : N 1 < m N }, D = {m : m N }\C and U = j C X j, V = j D X j. Then S N = U +V, CardC N and CardD = o( N ). From (2.1), it follows that for any δ > 0, P ( V δσ(2d N loglog N ) 1/2) <. (3.18) By the Borel-Cantelli lemma, we conclude that From (3.16) and (3.18), it follows that V = 0 a.s. (3.19) (2d N loglog N ) 1/2 P ( U (1 8ǫ)σ(2d N loglog N ) 1/2) =. (3.20) Note that {U ; N d } is a negatively associated filed. It follows that for any x,y and i j, P(U i x,u j y) P(U i x)p(u j y). Hence, by the generalized Borel-Cantelli lemma, (3.20) yields From (3.19) and (3.21), it follows that (3.15) holds. U (1 8ǫ)σ a.s. (3.21) (2d N loglog N ) 1/2 Proof of Theorem 1.2 is similar to that of (3.12) in which we choose m = 1 and Y = X instead. Proof of Theorem 1.3: From (1.6), it follows that there exists a constant 0 < C < such that ( ) S n P (2d n loglog n ) 1/2 < C > 0. 20

21 Then ( ) X n P < 2C > 0. (3.22) (2d n loglog n ) 1/2 Let A n = { X n 2C(2d n loglog n ) 1/2 }, A + n = {X+ n 2C(2d n loglog n )1/2 }, A n = {X n 2C(2d n loglog n ) 1/2 }. Note that { I{A + n };n Nd} and { I{A n };n Nd } are both negatively associated fields. It follows that for any m and n, It follows that { Var m m+n { Var { Var m m+n m m+n I{A + n } } } I{A n} } I{A } 2 m m+n m m+n m m+n P(A + n ), P(A n). P(A ). Hence from Lemma A.6 of Zhang and Wen (2001b), it follows that for any m ( 1 P( ma )) 2 m P(A ) 2P( m Since (3.22) implies P ( A n,i.o. ) < 1, we conclude that for some m, Then from (3.23), it follows that P( m P(A ) m A ) := β < 1. 2β (1 β 2 ) <. A ). (3.23) So, which implies P ( X 1 2C(2d loglog ) 1/2) <, m EX 2 1log d 1 ( X 1 )/loglog( X 1 ) <. (3.24) 21

22 Finally, from (3.24) and the law of the large numbers (c.f. Zhang and Wang 1999), it follows that lim n S n n EX 1 n 0 a.s. which together with (1.6) yields EX 1 = 0. And then (1.4) holds. References [1] Alam, K. and Saxena, K. M. L. (1981), Positive dependence in multivariate distributions, Comm. Statist. A 10: [2] Bulinsi, A. and Suquet, C. (2001), Normal approximation for quasi-associated random fields, Statist. Probab. Lett., 54: [3] Joag-Dev, K. and Proschan, F. (1983), Negative association of random variables, with applications, Ann Statist. 11: [4] Li, D. L and Wu, Z. Q. (1989), The law of the iterated logarithm for B-Valued random variables with multidimensional indices, Ann. Probab. 17: [5] Matula, P. (1992), A note on the almost sure convergence of sums of negatively dependent random variables. Statist. Probab. Lett. 15: [6] Móricz, F. (1983), A general moment inequality for the maximum of rectangular partial sums of multiple series, Acta. Math. Hungar. 41, [7] Newman, C. M. (1984), Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong, Y. L. (Ed.), Inequalities in Statistics and Probability. IMS, Hayward, CA., pp [8] Peligrad, M. and Gut, A. (1999), Almost sure results for a class of wea dependent random variables, J. Theor. Probab., 12:

23 [9] Roussas, C. G. (1996), Exponential probability inequalities with some applications, In Ferguson, T. S., Shapley, L. S. and MacQueen, J. B. (eds.), Statistics, Probability and Game Theory, IMS, Hayward, CA., pp [10] Shao, Q. M.(2000), A comparison theorem on maximum inequalities between negatively associated and independent random variables, J. Theor. Probab., 13: [11] Shao, Q. M. and Su, C. (1999), The law of the iterated logarithm for negatively associated random variables, Stochastic Processes & their Appl. 83: [12] Su, C., Zhao, L. C. and Wang, Y. B. (1997), The moment inequalities and wea convergence for negatively associated sequences, Science in China 40A: [13] Utev, S. and Peligrad, M. (2003), Maximal inequalities and an invariance principle for a class of wealy dependent random variables, J. Theor. Probab., 16: [14] Zhang, L. X. and Wang, X. Y. (1999), Convergence rates in the strong laws of asymptotically negatively associated random fields, Appl. Math.-JCU, Ser. B, 14(4): [15] Zhang, L. X. and Wen J. W. (2001b), A wea convergence for negatively associated fields, Statist. Probab. Lett., 53: [16] Zhang, L. X. and Wen, J. W. (2001a), Strong laws for sums of B-valued mixing random fields, Chinese Ann. Math., 22A: (in Chinese) Li-Xin Zhang Department of Mathematics Zhejiang University, Yuquan Campus Hangzhou , Zhejiang P.R. China stazlx@zju.edu.hz.cn 23