Convergence of weighted sums for dependent random variables
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(2) 884. Han-Ying Liang, Dong-Xia Zhang, and Jong-Il Baek. and only if n 1 α−1 An−k Xk convergences a.s. as n → ∞. Aαn k=0. It is well known that n 1 α−1 An−k Xk = µ a.s. n→∞ Aα n. lim. k=0. if and only if E|X|1/α < ∞ and EX = µ. For α = 1 this result is, of course, the classical Kolmogorove strong law. For 1/2 < α < 1 the proof is due to Lorentz ([14]); for 0 < α < 1/2 it follows from Chow and Lai ([4]). the case α = 1/2 was treated by D´eniel and Derriennic ([5]). Heinkel ([7]) established a version of this result in a Banach space setting. Li et al.([10]) studied the convergence rates of Ces`aro Law of Large Numbers and pointed out the following result. Theorem 1.1. Let {X, Xn , n ≥ 1} be a sequence of i.i.d. random variables. (i) For 0 < α < 1/2, if Eet|X| < ∞ for all t > 0, then n 1 α−1 An−k (Xk − EX) = o(n−α log n) a.s. Aαn k=0. (ii) For 1/2 < α < 1, if E(X − EX)2 = 1, then α(2α − 1)1/2 Γ2 (α)n1/2 (1/Aαn ). n k=0. D. Aα−1 n−k (Xk − EX) −→ N (0, 1).. However, many variables are dependent in actual problems. For example, negatively associated random variables, its definition is as follows: Definition. A finite family of random variables {Xi , 1 ≤ i ≤ n} is said to be negatively associated (NA) if for every pair of disjoint subsets A and B of {1, 2, . . . , n}, Cov(f1 (Xi , i ∈ A), f2 (Xj , j ∈ B)) ≤ 0 whenever f1 and f2 are coordinatewise increasing and such that the covariance exists. An infinite family of random variables is NA if every finite subfamily is NA. This definition is introduced by Alam and Saxena ([1]) and carefully studied by Joag-Dev and Proschan ([8]). As pointed out and proved by Joag-Dev and Proschan ([8]), a number of well known multivariate.
(3) Convergence of weighted sums for dependent random variables. 885. distributions possess the NA property, such as (a) multinomial, (b) convolution of unlike multinomials, (c) multivariate hypergeometric, (d) Dirichlet, (e) Dirichlet compound multinomial, (f) negatively correlated normal distribution, (g) permutation distribution, (h) random sampling without replacement, and (i) joint distribution of ranks. Perhaps, the significance of NA may lie, however, in the perception that NA is the appropriate modeling for several species competing for the same limited resources. Because of its wide applications in multivariate statistical analysis and systems reliability, the notion of NA have received considerable attention recently. we refer to Joag-Dev and Proschan ([8]) for fundamental properties, Matula ([15]) for the three series theorem, Su et al. ([23]) and Shao ([20]) for moment equalities, Shao and Su ([21]) for law of the iterated logarithm, Liang and Su ([11]) and Liang ([12]) for complete convergence, Roussas ([19]) for the central limit theorem of random fields. In addition, Kim and Baek ([9]) discussed a central limit theorem for linear processes generated by linearly positively quadrantdependent process. In order to extend Theorem 1.1 to NA setting, in this paper, we will discuss the strong convergence and central limit theorem for weighted sums of NA random variables. 2. Strong convergence Theorem 2.1. Let {Xni , 1 ≤ i ≤ kn , n ≥ 1} be an array of row NA random variables with EXni = 0 and P (|Xni | > x) = O(1)P (|X| > x) for all 1 ≤ i ≤ kn , n ≥ 1 and x > 0, which kn , n ≥ 1 is a sequence of positive integers. Assume that {ani , 1 ≤ i ≤ kn , n ≥ 1} is an array of real numbers satisfying max |ani | = O((log n)−1 ). (i) If. 1≤i≤kn. Eet|X| ∞ n=1. (ii). kn . a2ni = o((log n)−1 ).. i=1. < ∞ for all t > 0, then. n. r−2. P (|. kn . ani Xni | > ) < ∞ for all > 0 and all r ≥ 2.. i=1. Remark 2.1. The following example shows that Theorem 2.1 does not hold if the condition (ii) is replaced by the weaker condition (ii)’ kn 2 −1 i=1 ani = O((log n) )..
(4) 886. Han-Ying Liang, Dong-Xia Zhang, and Jong-Il Baek. Example 1. Let {X, Xi , i ≥ 1} be a sequence of iid N (0, 1) random variables. Set 1/[log n], if 1 ≤ i ≤ [log n], ani = 0, if [log n] + 1 ≤ i ≤ n, where [x] denotes the integer part of x. Then the condition (i) of Theorem 2.1 and the above condition (ii)’ are easily satisfied. 2 Note that X ∼ N (0, 1), it follows that Eet|X| ≤ 2et /2 for all t > 0. [log n] Since i=1 Xi / [log n] ∼ N (0, 1), we have by Lemma 5.1.1 in Stout ([22]) that P (|. n . [log n]. ani Xi | > 1) = P (|. i=1. . Xi / [log n] | > [log n]). i=1. ≥ 2 exp{−[log n]} ≥ 2/n for all sufficiently large n, and so ∞ n nr−2 P (| ani Xi | > 1) = ∞ for all r ≥ 2, n=1. i=1. i.e. Theorem 2.1 does not hold. Corollary 2.1. Let {Xi , i ≥ 0} be a sequence of NA random variables and P (|Xi | > x) = O(1)P (|X| > x) for all i ≥ 0 and x > 0. For 0 < α < 1/2, if Eet|X| < ∞ for all t > 0, then n 1 α−1 An−k (Xk − EXk ) = o(n−α log n) a.s. Aαn k=0. 3. Central limit theorem Theorem 3.1. Let {Xi , −∞ < i < ∞} be a sequence of mean zero NA random variables satisfying |cov(Xk , Xj )| → 0 as u → ∞ uniformly for k ≥ 1. (3.1) j:|k−j|≥u. Assume that {ani , 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers satisfying n a2ni = O(1) and max |ani | → 0 as n → ∞ (3.2) i=1. and that Var( ni=1 ani Xi ) → 1.. 1≤i≤n.
(5) Convergence of weighted sums for dependent random variables. 887. (a) If Xi is uniformly integrable in L2 , then n . D. ani Xi −→ N (0, 1),. i=1. (b) Put ξt = ∞ c X . Here {cj } is a sequence of real numbers ∞ j=0 j t−j ∞ with C(1) = c j=0 j = 0 and j=1 j|cj | < ∞. If Xi is uniformly integrable in L2 and max1≤i≤n |ani | = O(n−1/2 ), then n . D. ani ξi −→ N (0, C 2 (1)),. i=1. (c) Put ηt = ∞ c X . Here {c } is a sequence of real numbers ∞ j=−∞ j t−j ∞ j 2 2 with D(1) = j=−∞ cj = 0 and j=−∞ j cj < ∞. If supi E|Xi |2+δ < ∞ for any δ > 0 and max1≤i≤n |ani | = O(n−1/2 ), then n . D. ani ηi −→ N (0, D2 (1)).. i=1. Corollary 3.1. Let {Xi , i ≥ 1} be a sequence of NA random variables. For 1/2 < α < 1, if Xi is uniformly integrable in L2 , then Rn = α(2α − 1)1/2 Γ2 (α)n1/2 (1/Aαn ). n k=0. D. 2 Aα−1 n−k (Xk − EX) −→ N (0, σ ),. where σ 2 = limn→∞ VarRn . Remark 3.1. In Corollary 3.1, if {X, Xn , n ≥ 1} is a sequence of i.i.d. random variables and E(X − EX)2 = 1, then σ 2 = 1. Since independent r.v.’s are a special case of NA r.v.’s, Corollarys 2.1 and 3.1 extend Theorem 1.1 to NA case.. 4. Proofs of Theorems 2.1 and 3.1 In this section, a b means a = O(b), a+ = max(0, a), a− = max(0, −a). Let C and c denote positive constant whose values are unimportant and may vary at different place..
(6) 888. Han-Ying Liang, Dong-Xia Zhang, and Jong-Il Baek − Proof of Theorem 2.1. Since ani = a+ ni − ani , it suffices to show. (4.3). ∞ . n. r−2. P (|. n=1. (4.4). ∞ . nr−2 P (|. n=1. kn i=1 kn i=1. a+ ni Xni | > ) < ∞. for any > 0,. a− ni Xni | > ) < ∞. for any > 0.. We prove only (4.3), the proof of (4.4) is analogous. To prove (4.3), we need only to prove (4.5). ∞ n=1. (4.6). ∞ . kn nr−2 P ( a+ ni Xni > ) < ∞ i=1 kn . nr−2 P (. n=1. i=1. for any > 0,. a+ ni Xni < −) < ∞ for any > 0.. We first prove (4.5). From the definition of NA variables, we know that {a+ ni Xni , 1 ≤ i ≤ kn , n ≥ 1} is still an arrays of row NA random variables. Noticing ex ≤ 1 + x + 12 x2 e|x| for all x ∈ R, hence by using lemma 1 of Matula ([15]), we get for t = M log n/, where M is a large constant and will be specified later on, ∞ . ≤. kn nr−2 P ( a+ ni Xni > ). n=1 ∞ . i=1. nr−2 e−t Eet. kn. i=1. a+ ni Xni. n=1. ≤. ∞ . nr−2−M. n=1. ≤. ∞ n=1. . ∞ n=1. ≤. ∞ n=1. kn . +. Eetani Xni. i=1 kn 1 2 2 ta+ ni |Xni | ] nr−2−M [1 + t2 (a+ ni ) EXni e 2. nr−2−M. i=1 kn i=1. 2 (1+C)|X| [1 + C(log n)2 (a+ ] ni ) Ee. nr−2−M exp{C(log n)2. kn 2 (a+ ni ) } i=1.
(7) Convergence of weighted sums for dependent random variables. ≤. ∞ . 889. n(r+)−(2+M ) < ∞. n=1. provided M > (r + ) − 1. Thus, (4.5) is proved. By replacing Xni by −Xni from the above statement, and noticing + {ani (−Xni ), 1 ≤ i ≤ kn , n ≥ 1} is still an arrays of row NA random variables, we know that (4.6) holds. Lemma 1. (Cai and Roussas ([3])) Let A and B be disjoint subsets of N , and let {Xj , j ∈ A ∪ B} be NA. (i) Let f : R#A → R and g : R#B → R be partially differentiable with bounded partial derivatives, and let ∂f /∂ti ∞ stand for the supnorm. Then |Cov{f (Xi ; i ∈ A), g(Xj ; j ∈ B)}| ∂f ∂g. ∞ ·. ∞ |Cov(Xi , Xj )|. ≤ ∂ti ∂tj i∈A j∈B. (ii) If q : R → R is a bounded differentiable function with bounded derivative, then |Cov{. . q(Xi ),. i∈A. . q(Xj )}| ≤ q #A+#B−2. q 2∞ ∞. j∈B. . |Cov(Xi , Xj )|.. i∈A j∈B. Proof of Theorem 3.1. (a) This part has been proved partially in Liang and Jing ([13]), but, for the sake of easy reference, we give a proof here, too. Without loss of generality, we assume that ani = 0 for all i > n. Note that, for 1 ≤ u ≤ n − 1 n . |ani anj Cov(Xi , Xj )| ≤ sup | k. i,j=1,|i−j|≥u. . n Cov(Xi , Xj )|( a2ni ),. j,|k−j|≥u. i=1. and hence, by (3.1) and (3.2), for a fixed small > 0, we can find a positive integer u = u such that 0≤. i,j=1,|i−j|≥u. |ani anj Cov(Xi , Xj )| ≤ ..
(8) 890. Han-Ying Liang, Dong-Xia Zhang, and Jong-Il Baek. Denote by [x] the integer part of x and define 1 K = [ ], u(j+1) ani Xi , j = 0, 1, 2, . . . Ynj = i=uj+1. Aj = {i : 2Kj ≤ i < 2Kj + K, |Cov(Yni , Yn,i+1 )| ≤. 2 K. 2Kj+K . V ar(Yni )}.. i=2Kj. Since 2|Cov(Yni , Yn,i+1 )| ≤ V ar(Yni )+V ar(Yn,i+1 ), we get that for every j the set Aj is not empty. Now we define the integers m1 , m2 , . . . , mn recurrently by m0 = 0: mj+1 = min{m : m > mj , m ∈ Aj } and put mj+1. . Znj =. Yni , j = 0, 1, 2, . . .. i=mj +1. j = {u(mj + 1) + 1, . . . , u(mj+1 + 1)}. We observe that. . Znj =. ank Xk , j = 0, 1, . . .. k∈j. It is easy to see that every set j contains no more than 3Ku elements. Thus, by (3.2), we know that the uniformly integration of {Xi2 , i ≥ 1} implies the uniformly integration of {Zni , 1 ≤ i ≤ n, n ≥ 1}, and hence {Zni , 1 ≤ i ≤ n, n ≥ 1} satisfies the Lindeberg’s Condition. It remains to observe that by Lemma 1, for any real number t |E exp(it. n . Znj ) −. j=1. ≤ |Cov(exp(it. n−1 . n . E exp(itZnj )|. j=1. Znj ), exp(itZnn ))|. j=1. +|E exp(itZnn )||E exp(it. n−1 j=1. Znj ) −. n−1 j=1. E exp(itZnj )|.
(9) Convergence of weighted sums for dependent random variables. . t2. |Cov(Zni , Znj )|. 1≤i<j≤n. =. . t2 [. . t [. +. . |Cov(Zni , Znj )| +. 1≤i<j≤n,|i−j|=1 2. 891. |Cov(Zni , Znj )|]. 1≤i<j≤n,|i−j|>1. |ani anj ||Cov(Xi , Xj )|. 1≤i<j≤n,|i−j|≥u n . |Cov(Ynmj , Yn,mj+1 )|]. j=1. t2 [ +. n. O(1) V ar(Yni )] t2 . K i=1. Now, Theorem 3.1(a) is proved by Theorem 4.2 in Billingsley ([2]). (b) Note that ∼. ∼. where X k = n . ∞. j=0. ∼. ξk = C(1)Xk + X k−1 − X k , ∼ ∼ c j Xk−j and c j = ∞ i=j+1 ci . Hence. ani ξi = C(1). i=1. n . ank Xk +. k=1. n . ∼. ∼. ank (X k−1 − X k ) := In + Jn .. k=1. D. By (a), we get In −→ N (0, C 2 (1)). P. To prove Jn → 0, we here state Abel Inequality (see p.32, Theorem 1 of Mitrinovic ([16])): Let A1 , A2 , . . . , An ; B1 , B2 , . . . , Bn (B1 ≥ B2 ≥ · · · ≥ Bn ≥ 0) be two sequences of real numbers, and let Sk = ki=1 Ai , M1 = min1≤k≤n Sk and M2 = max1≤k≤n Sk . Then B1 M1 ≤. (4.7). n . Ak Bk ≤ B1 M2 .. k=1. Without loss of generality, assume that an1 ≥ an2 ≥ · · · ≥ ann . Let Bs = ans − ann , 1 ≤ s ≤ n − 1, Bn = 0. Applying (4.7) we have |Jn | ≤ |. n . ∼. ∼. (ank − ann )(X k−1 − X k )| + |. k=1. ≤ 2 max |ank − ann | max | 1≤k≤n. 1≤m≤n. m . n . ∼. k=1 ∼. ∼. (X k−1 − X k )|. k=1. ∼. ann (X k−1 − X k )|.
(10) 892. Han-Ying Liang, Dong-Xia Zhang, and Jong-Il Baek ∼. ∼. +|ann || X 0 − X n | ∼. (4.8). ∼. max |ank (x)|(| X 0 | + max | X m |). 1≤k≤n. O(n. =. −1/2. ∼. )(| X 0 | +. 1≤m≤n ∼ max | X m 1≤m≤n. |).. ∞ ∼ Since ∞ j=1 j|cj | < ∞ ⇒ j=0 | c j | < ∞ (see Phillips and Solo ([18]), Lemma 2.1), ∼. E| X 0 | ≤. (4.9). ∞ . ∼. | c j |E|X−j | < ∞.. j=0. On the other hand, observe that m . ∼. | Xm | ≤. ∼. | c i ||Xm−i | +. i=0. ∧. with c i =. ∞. j=i+1 |cj |. ∞ . (4.11). ∼. | c m+i ||X−i |. i=1. m ∞ ∼ ∧ | c i |) + | c i ||X−i | max |Xi |(. ≤. (4.10). ∞ . 0≤i≤m. i=0. i=1. Note that. ∼. | cj | ≤. j=1. ∞ ∞ ∞ ∞ ∧ cj = |ci | ≤ j|cj | < ∞, j=1. j=1 i=j+1. j=1. P. and n−1/2 max0≤m≤n |Xm | → 0 is equivalent to n−1. n m=0. P. 2 Xm I(|Xm | > n1/2 ) → 0,. ∀ > 0. (cf. Hall and Heyde ([6]), p.53), which, together with (4.8)-(4.11), folP. lows Jn → 0. (c) Note that i−1 − X i + X ηi = D(1)Xi + X i+1 − X i , 0 cj Xi−j and i = ∞ cj = ∞ where X j=0 cj Xi−j , X i = j=−∞ k=j+1 ck , j−1 ci . cj = i=−∞. Similarly to the proof in (b), we need only prove that 0 |2 = op (1), n−1 |X. m |2 = op (1), n−1 max |X 1≤m≤n.
(11) Convergence of weighted sums for dependent random variables. 2 n−1 |X 1 | = op (1), ∞. 893. 2 n−1 max |X m | = op (1). 1≤m≤n. 2 0 |2 < ∞ and E|X < ∞, we can get E|X 1 | < ∞, which 0 |2 = op (1) and n−1 |X 1 |2 = op (1), respectively. follow n−1 |X On the other hand, note that m |2 = op (1) if and only if n−1 max |X. By. (4.12). j=−∞ j. 2 c2 j. 1 n. 1≤m≤n n i2 I(X i2 X i=1. > nc) = op (1) for any c > 0.. 2 n−1 max |X m | = op (1) if and only if (4.13). 1 n. 1≤m≤n n 2. 2 X i I(X i > nc) = op (1) for any c > 0. i=1 2. 2 } and {X (cf. Hall and Heyde ([6]), p.53). Since {X i } are uniformly i 2 c2 < ∞ and sup E|X |2+δ < ∞, by (4.12) and integrable by ∞ j i i j j=−∞ (4.13) we get 2 m |2 = op (1), n−1 max |X n−1 max |X m | = op (1). 1≤m≤n. 1≤m≤n. References [1] K. Alam and K. M. L. Saxena, Positive dependence in multivariate distributions, Comm. Statist. Theory Methods A10 (1981), 1183–1196. [2] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. [3] Z. W. Cai and G. G. Roussas, Berry-Esseen bounds for smooth estimator of distribution function under association, Nonparametric Statist. 11 (1999), 79–106. [4] Y. S. Chow and T. L. Lai, Limiting behavior of weighted sums of independent random variables, Ann. Probab. 1 (1973), 810–824. [5] Y. D´eniel and Y. Derriennic, Sur la convergence presque sure, au sens de Ces` aro d’ordre α, 0 < α < 1, de variables al´eatoires et ind´ependantes et identiquement distribu´ees, Probab. Theory Related Fields 79 (1988), 629–636. [6] P. Hall and C. C. Heyde, Martingale Limit Theory and Its Applications, New York, Academic Press, 1980. [7] B. Heinkel, An infinite-dimensional law of large numbers in Ces` aro’s sense, J. Theoret. Probab. 3 (1990), 533–546. [8] K. Joag-Dev and F. Proschan, Negative association of random variables with applications, Ann. Statist. 11 (1983), 286–295. [9] T.-S. Kim and J.-I. Baek, A central limit theorem for stationary linear processes generated by linearly positively quadrant dependent process, Statist. Probab. Lett. 51 (2001), 299–305..
(12) 894. Han-Ying Liang, Dong-Xia Zhang, and Jong-Il Baek. [10] D. L. Li, M. B. Rao, T. F. Jiang and X. C. Wang, Complete convergence and almost sure conver-gence of weighted sums of random variables, J. Theoret. Probab. 8 (1995), 49–76. [11] H. Y. Liang and C. Su, Complete convergence for weighted sums of NA sequences, Statist. Probab. Lett. 45 (1999), 85–95. [12] H. Y. Liang, Complete convergence for weighted sums of negatively associated random variables, Statist. Probab. Lett. 48 (2000), 317–325. [13] H. Y. Liang and B. Y. Jing, Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences, Submitted, 2002. [14] G. G. Lorentz, Borel and Banach properties of methods of summation, Duke Math. J. 22 (1955), 129–141. [15] P. Matula, A note on the almost sure convergence of sums of negatively dependences random variables, Statist. Probab. Lett. 15 (1992), 209–213. [16] D. S. Mitrinovic, Analytic Inequalities, New York, Springer, 1970. [17] M. Peligrad and S. Utev, Central limit theorem for linear processes, Ann. Probab. 25 (1997), no. 1, 443–456. [18] P. C. B. Phillips and V. Solo, Asymptotics for linear processes, Ann. Statist. 20 (1992), 971–1001. [19] G. G. Roussas, Asymptotic normality of random fields of positively or negatively associated processes, J. Multivariate Anal. 50 (1994), 152–173. [20] Q. M. Shao, A comparison theorem on maximum inequalities between negatively associated and independent random variables, J. Theoret. Probab. 13 (2000), 343– 356. [21] Q. M. Shao and C. Su, The law of the iterated logarithm for negatively associated random variables, Stochastic Process. Appl. 83 (1999), 139–148. [22] W. F. Stout, Almost Sure Convergence, Academic Press, New York, 1974. [23] C. Su, L. C. Zhao and Y. B. Wang, Moment inequalities and week convergence for negatively associated sequences, Sci. China Ser. A 40 (1997), 172–182.. Han-Ying Liang and Dong-Xia Zhang Department of Applied Mathematics Tongji University Shanghai 200092, P. R. China E-mail : [email protected] Jong-Il Baek School of Mathematics & Informational Statistics and Institute of Basic Natural Science Wonkwang University Ik-San 570-749, Korea E-mail : [email protected].
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