MOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES

J. Korean Math. Soc. 47 1, No., pp. 63 75 DOI 1.4134/JKMS.1.47..63 MOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES Ke-Ang Fu Li-Hua Hu Abstract. Let X n ; n 1 be a strictly stationary sequence of negatively associated rom variables with mean zero finite variance. Set S n = n k=1 X k, M n = max k n S k, n 1. Suppose σ = EX1 k= EX 1X k < σ <. We prove that for any b > 1/, if E X δ < δ 1, then log log n b 1/ ε εb1 = 1/ b σe N b1 b 1b 1 for any b > 1/, = ε ε b1 Γb 1/ b 1 n 3/ log n k 1 b1 E M n σε 1 k log log n b n 3/ log n E σε 1 k k 1 b, π n 8 log log n M n where Γ is the Gamma function N sts for the stard normal rom variable. 1. Introduction Let X n ; n 1 be a sequence of rom variables with common distribution, EX 1 = < EX1 <. Set S n = n k=1 X k, M n = max k n S k, n 1 denote log x = lnx e, log log x = loglog x. When X n ; n 1 is a sequence of i.i.d. rom variables, Gut Spǎtaru [4] discussed the convergence rates of the usual law of the iterated logarithm, proved the following theorem. Received May 11, 8; Revised July, 8. Mathematics Subject Classification. 6F15, 6G5. Key words phrases. Chung-type law of the iterated logarithm, moment convergence rates, negative association, the law of the iterated logarithm. Project supported by National Natural Science Foundation of ChinaNos. 1671176, 177119 & 787113. 63 c 1 The Korean Mathematical Society

64 KE-ANG FU AND LI-HUA HU Theorem A. Suppose that EX 1 = < EX1 <. Then we have 1 ε ε P S n ε n log log n = EX1. Also Chow [3] first discussed the moment convergence of i.i.d. rom variables, got the following result. Theorem B. Suppose that EX =. Assume p 1, α > 1/, pα > 1 E X p X log1 X <. Then for any ε >, n pα α E max S j εn α <. j n Recently, Jiang Zhang [5] obtained the precise rates in the law of the iterated logarithm for the moment convergence of i.i.d. rom variables via strong approximation method. In this paper, we consider the moment convergence rates in the usual law of the iterated logarithm the Chung-type law of the iterated logarithm for negatively associated NA rom variables. Notice that the proof pattern is similar with that of Gut Spǎtaru [4], our emphasis is on the convergence rates of M n. Also we obtained the theorems without the help of the Berry-Esseen theorem which was used in many other cases c.f. Li [7]. First, we shall give the definition of negatively associated rom variables: Definition 1. A finite sequence of rom variables X k ; 1 k n is said to be negatively associated NA, if for every disjoint subsets A B of 1,,..., n, we have CovfX i ; i A, gx j ; j B, whenever f g are coordinatewise increasing the covariance exists. An infinite sequence of rom variables is NA if every finite subsequence is NA. The notion of NA was first introduced by Alam Saxena [1]. Joag-Dev Proschan [6] showed that many well known multivariate distributions possess the NA property. Because of its wide application in multivariate statistical analysis system reliability, the notion of NA has received considerable attention recently. We refer to Joag-Dev Proschan [6] for fundamental properties, Shao [8] for the moment inequalities for partial sums maximum of partial sums, Shao Su [9] for the law of the iterated logarithm, which holds under finite variance. The content of the paper is organized as follows. In Section we list the main results. The proofs of Theorems.1. are given in Sections 3 4, respectively. In what follows let M C etc. denote positive constants whose values possibly vary from place to place. The notation of a n b n means that an b n 1 as n, [x] denotes the largest integer less than x.

MOMENT CONVERGENCE RATES OF LIL 65. Main results Here in the sequel, let X n ; n 1 be a sequence of strictly stationary NA rom variables, EX 1 =, < EX1 <, < σ := EX1 k= EX 1X k < < σ < unless it is specially mentioned. Now we are in a position to present our main results. Theorem.1. For any b > 1/, if E X δ < < δ 1, then we have log log n b 1/ ε εb1 E M n 3/ n σε log n = 1/ b σe N b1 1 k b 1b 1 k 1 b1.1 = ε εb1 1/ b σ b 1b 1 E N b1, log log n b 1/ E S n 3/ n σε log n where N is the stard normal rom variable. Remark.1. The theorem holds under finite δth moment since the central it theorem weak invariance principle for NA sequences are applied, while it holds under finite variance for i.i.d. case. Also here we obtain the moment convergence rates for M n. Theorem.. For any b > 1/, we have = ε ε b1 Γb 1/ b 1 log log n b n 3/ log n E σε 1 k k 1 b, where Γ is the Gamma function. π n 8 log log n M n Remark.. It is surprising that the precise rates in the Chung-type law of the iterated logarithm holds under finite variance, since we avoid the use of the Berry-Esseen theorem. 3. The proof of Theorem.1 In this section, we shall present the proof of Theorem.1. First, we give some lemmas which will be used in the following proofs.

66 KE-ANG FU AND LI-HUA HU Lemma 3.1 []. Let W t; t be a stard Wiener process, let N be a stard normal rom variable. Then for any x > P sup W s x = 1 1 k Pk 1x N k 1x k= = 4 1 k PN k 1x In particular, P sup W s x = 1 k P N k 1x. PN x πx exp k=1 x as x. Also, for any x >, P sup W s x = 4 1 k π k 1 exp π k 1 8x P sup W s x 4π exp π 8x as x. Lemma 3. [11]. Under the moment condition of Theorem.1, we have M n σ n sup W s S n σ N in distribution. n Lemma 3.3 [8]. Let Y i ; 1 i n be a sequence of NA rom variables with mean zero finite variance. Denote S k = k i=1 Y i, 1 k n, B n = n i=1 EY i, then for any z >, y >, z/1y P max S k z P max Y k y 4 exp z B n 4. k n k n 8B n 4zy B n Now set bε = expexpm/ε, M > 4 < ε < 1/4, say. Without loss of generality, assume σ = 1. Lemma 3.4. For any M > 4 b > 1/, we have log log n b 1/ ε εb1 n 1/ E M n ε n bε E sup W s ε 3.1 log log n =

MOMENT CONVERGENCE RATES OF LIL 67 3. ε εb1 n bε log log n b 1/ n 1/ E S n ε E N ε log log n =. Proof. We only give the proof of 3.1, since the proof of 3. is similar. Note that ε b1 Cε b1 n bε n bε log log n b 1/ n 1/ E M n ε log log n b Γn Set = log log n 1/ 1/ n n = sup x P E sup W s ε log log n PM n x n P PM n x ε sup W s x ε log log n dx. sup W s x, from Lemma 3., it follows that n as n. Thus via Toeplitz Lemma [1], it immediately leads to 3.3 ε b1 ε b1 n bε n bε log log n b Γn 1/ n log log n b 1/ M b1/ 1 log log[bε] b1/ M P n x ε P sup W s x ε log log n dx n bε 1/ n log log n b 1/ as ε.

68 KE-ANG FU AND LI-HUA HU Also observe that ε b1 n bε =: ε b1 n bε P M n x ε log log n b P sup W s x ε log log n dx log log n b II 1 II dx. Now we deal with II 1 II, respectively. For II 1, denote θ = 1/EX1, therefore by using Lemma 3.3 where we take z = xε, y = βz B n = n/θ, it leads to II 1 dx np X 1 βx ε 4 exp 4 Clog log n 1 n/θ 8βx ε n log log n 1/1β dx x ε dx Clog log n 1/ n 1/. θ x ε log log n 4 As to II, we have that, by Lemma 3.1, for any m 1 x >, m1 1 k P N k 1x P sup W s x 3.4 Thus it provides that II dx = m m P m 1 k P N k 1x. sup W s x ε log log n 1 k dx P N k 1x ε log log n dx 1 k log log n 1 k 1 Clog log n 1/ n 1/. x ε dx Then an application of Toeplitz Lemma [1] again, combined with 3.3, completes the proof.

MOMENT CONVERGENCE RATES OF LIL 69 Lemma 3.5. For < ε < 1/4 b > 1/, we have uniformly log log n b 1/ M εb1 E sup W s ε log log n n>bε =. Proof. For k large enough, we obtain ε b1 log log n b 1/ P sup W s ε log log n x dx ε b1 Cε b1 Cε b1 n>bε n>bε n>bε n>bε log log n b log log n b P N x ε log log n dx E N k dx x ε k log log n k/ log log n b k/ ε k1 = CM b1 k/, when M, uniformly for < ε < 1/4. Lemma 3.6. For b > 1/, if E X 1 δ < < δ 1, then M ε εb1 n>bε log log n b 1/ E M n 3/ n ε =. log n Proof. Write θ = 1/EX 1. By applying Lemma 3.3 where z = x ε, y = β z B n = n/θ again, similarly with Lemma 3.4, we have ε b1 n>bε Cε b M 1 δ/ log log n b 1/ E M n 3/ n ε log n n>bε 1 n 1δ/ C θ M/4 s b e s ds Cε 1 log log bε. Then we can get the desired result immediately by letting ε M. Proposition 3.1. For any b > 1/, we have log log n b 1/ ε εb1 E sup W s ε 3.5 log log n = 1/ b E N b1 1 k b 1b 1 k 1 b1

7 KE-ANG FU AND LI-HUA HU 3.6 ε εb1 log log n b 1/ E N ε log log n where N is the stard normal rom variable. = 1/ b E N b1, b 1b 1 Proof. Note that for any m 1 x >, 3.4 holds. Thus it follows that for any t > E sup W s t = P sup W s t x dx 3.7 = = E sup W s t m m m 1 k P N k 1t x dx 1 k k 1 P N k 1t x dx 1 k k 1 E N k 1t m1 1 k k 1 E N k 1t. So, it suffices to show that for any q 1 b > 1/, ε εb1 Obviously, 3.8 log log n b 1/ ε εb1 = ε ε b1 E N qε log log n log log n b 1/ e e = 1/ b q b 1 ε log log y b 1/ y log y = q b 1 1/ b E N b1. b 1b 1 = q b 1 1/ b E N b1. b 1b 1 E N qε log log n qε log log y z b P N xdxdz qε z P N xdxdy Thus the proposition is now proved by taking q = k1 q = 1, respectively.

MOMENT CONVERGENCE RATES OF LIL 71 Proposition 3.. For any b > 1/, we have ε εb1 log log n b 1/ n 1/ E E M n ε sup W s ε log log n = ε εb1 log log n b 1/ n 1/ E S n ε E N ε log log n =. Proof. It is trivial from Lemmas 3.4-3.6. The Proof of Theorem.1. By using the Propositions 3.1 3., we can easily get the conclusions. 4. The proof of Theorem. Similarly, we state some lemmas before showing the proof of Theorem.. Lemma 4.1. For any b > 1/ M > 4, we have ε ε b1 n b1/ε log log n b n 1/ E ε π n 8 log log n M n π E ε 8 log log n sup W s =. Proof. Take n = sup x PM n x n Psup W s x here, thus via Toeplitz Lemma [1], we get the desired result by following the lines in Lemma 3.4. Lemma 4.. For ε > sufficiently large b > 1/, we have M ε b1 uniformly in ε. n>b1/ε log log n b E π ε 8 log log n sup W s =,

7 KE-ANG FU AND LI-HUA HU Proof. By virtue of Lemma 3.1, we have that ε b1 log log n b E π ε 8 log log n sup W s n>b1/ε = ε b1 C M n>b1/ε log log n b ε π 8 log log n y b 1/ e y dy as M. P sup W s t dt Lemma 4.3. There exist constants λ > C > such that for any x 1 n 1, PM n x λ log log n n/ log log n C exp x. Proof. It follows from Lemma 3. that PS m m PN as m. Thus, there exist λ > m such that PS m m e λ < 1, when m m /. Set m = [nx / log log n] N = [n/m1]. Thus, when x 1 n m, we have m m /. Notice that S km1 S k 1m1, k = 1,..., N, are also negatively associated. Hence, for those x 1 n m satisfying x / log log n < 1/4, we have M n x P M n m 1 n P log log n P S km1 S k 1m1 m 1; k = 1,,..., N P S km1 S k 1m1 m 1; k = 1,,..., N N P S km1 S k 1m1 m 1 k=1 = P N S m1 m 1 e λn n n exp λ m 1 1 exp λ m 1 log log n exp λ x 1 e λ exp On the other h, if x / log log n > 1/4, then P M n x n/ log log n 1 e λ exp λ log log n x. λ log log n x. Moreover, when x 1 n m, we have P M n x n/ log log n 1 e λ log log λ log log n m exp x.

MOMENT CONVERGENCE RATES OF LIL 73 Hence, the conclusion follows, as desired. Lemma 4.4. Uniformly for ε sufficiently large, we have for b > 1/, M ε b1 n>b1/ε log log n b n 3/ log n E π ε n 8 log log n M n =. Proof. Notice that for any ε large enough, it follows from Lemma 4.3 that ε b1 log log n b n 3/ log n E π ε n 8 log log n M n Cε b 1 Cε b 1 C M n>b1/ε n>b1/ε n>b1/ε y b 1/ exp log log n b 1/ log log n b 1/ P M n ε π n 8 log log n 4λ log log n exp ε π 4λ π y dy as M. Proposition 4.1. For any b > 1/, we have = ε ε b1 Γb 1/ b 1 log log n b E ε 1 k k 1 b. π 8 log log n sup W s Proof. It follows from Lemma 3.1 that log log n b ε ε b1 E π ε 8 log log n sup W s = ε ε b1 = ε ε b1 = 4 π log log n b log log n b ε π 8 log log n ε π 8 log log n 1 k k 1 log log x b ε ε b1 e x log x e P 4 π ε π 8 log log x sup W s t 1 k k 1 exp dt π k 1 8t exp π k 1 8t dt dtdx

74 KE-ANG FU AND LI-HUA HU = 1 1 k ε ε b1 e e log log x b x log x = 1 1 k s b k 1 b ε k1 /ε = 1 1 k k 1 b ε = = 1 1 k b 1 k 1 b ε Γb 1/ b 1 s k1 /ε y 3/ e y y 3/ e y dydx k1 log log x/ε πk 1 t = y 1/ y 3/ e y dyds y k1 /ε s b dsdy k1 /ε y b 1/ e y dy 1 k k 1 b. Proposition 4.. For any b > 1/, we have sup ε b1 log log n b nπ ε n 1/ E εσ 8 log log n M n E ε Proof. From Lemmas 4.1, 4. 4.4, it follows easily. π 8 log log n sup W s =. The Proof of Theorem.. By virtue of Propositions 4.1 4., we complete the proof immediately. Acknowledgements. The authors thank the referee for several comments that have led to improvements in this work. References [1] K. Alam K. M. L. Saxena, Positive dependence in multivariate distributions, Comm. Statist. A Theory Methods 1 1981, no. 1, 1183 1196. [] P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York- London-Sydney, 1968. [3] Y. S. Chow, On the rate of moment convergence of sample sums extremes, Bull. Inst. Math. Acad. Sinica 16 1988, no. 3, 177 1. [4] A. Gut A. Spătaru, Precise asymptotics in the law of the iterated logarithm, Ann. Probab. 8, no. 4, 187 1883. [5] Y. Jiang L. X. Zhang, Precise rates in the law of iterated logarithm for the moment of i.i.d. rom variables, Acta Math. Sin. Engl. Ser. 6, no. 3, 781 79. [6] K. Joag-Dev F. Proschan, Negative association of rom variables, with applications, Ann. Statist. 11 1983, no. 1, 86 95. [7] Y. X. Li, Precise asymptotics in complete moment convergence of moving-average processes, Statist. Probab. Lett. 76 6, no. 13, 135 1315.

MOMENT CONVERGENCE RATES OF LIL 75 [8] Q. M. Shao, A comparison theorem on moment inequalities between negatively associated independent rom variables, J. Theoret. Probab. 13, no., 343 356. [9] Q. M. Shao C. Su, The law of the iterated logarithm for negatively associated rom variables, Stochastic Process. Appl. 83 1999, no. 1, 139 148. [1] W. F. Stout, Almost Sure Convergence, Academic, New-York, 1995. [11] C. Su, L. C. Zhao, Y. B. Wang, The moment inequalities weak convergence for negatively associated sequences, Sci. China Ser. A 4 1997, no., 17 18. Ke-Ang Fu College of Statistics Mathematics Zhejiang Gongshang University Hangzhou 3118, P. R. China E-mail address: statzju@tom.com Li-Hua Hu College of Statistics Mathematics Zhejiang Gongshang University Hangzhou 3118, P. R. China E-mail address: lihuahulihua@16.com