Almost Sure Central Limit Theorem for Self-Normalized Partial Sums of Negatively Associated Random Variables

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1 Filomat 3:5 (207), DOI /FIL70543W Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Almost Sure Central Limit Theorem for Self-Normalized Partial Sums of Negatively Associated Random Variables Qunying Wu a, Yuanying Jiang a, a College of Science, Guilin University of Technology, Guilin 54004, P.R. China Abstract. Let X, X, X 2,... be a stationary sequence of negatively associated random variables. A universal result in almost sure central it theorem for the self-normalized partial sums S n /V n is established, where: S n = n i= X i, Vn 2 = n i= X 2. i. Introduction Starting with Brosamler [] and Schatte [2], several authors investigated the almost sure central it theorem (ASCLT) for partial sums S n /σ n of random variables in the last two decades. We refer the reader to Brosamler [], Schatte [2], Lacey and Philipp [3], Ibragimov and Lifshits [4], Beres and Csái [5], Hörmann [6], Miao [7] and Wu [8] in this context. If σ n is replaced by an estimate from the given data, n usually denoted by V n = i= X2 i, V n is called a self-normalizer of partial sums. A class of self-normalized random sequences has been proposed and studied in Peligrad and Shao [9], Pena et al. [0] and references therein. The past decade has witnessed a significant development in the field of it theorems for the self-normalized sum S n /V n. We refer to Bentus and Gótze [] for the Berry-Esseen bound, Gine et al. [2] for the asymptotic normality, Hu et al. [3] for the Cramer type moderate deviations, Csörgo et al. [4] for the Donser s theorem, Huang and Pang [5], Zhang and Yang [6] and Wu [7] for the almost sure central it theorems. In addition, Wu [7] proved the ASCLT for the self-normalized partial sums that reads as follows: Let {X, X n } n N be a sequence of i.i.d. random variables in the domain of attraction of the normal law with mean zero. Then = { } S d I x = Φ(x) a.s. for any x R, V where: d = exp(lnα ), = n = d, 0 α < /2, I denotes indicator function, and Φ(x) stands for the standard normal distribution function. 200 Mathematics Subect Classification. Primary 60F5 Keywords. Negatively associated random variables; Self-normalized partial sums; Almost sure central it theorem. Received: 5 February 205; Revised 3 October 205; Accepted: 27 March 206 Communicated by Miroslav M. Ristić Research supported by the National Natural Science Foundation of China (3609, 66029) and the Support Program of the Guangxi China Science Foundation (204GXNSFAA805, 205GXNSFAA39008) Corresponding author addresses: wqy666@glut.edu.cn (Qunying Wu), yy@glut.edu.cn (Yuanying Jiang)

2 Wu and Jiang / Filomat 3:5 (207), Many results concerning the it theory for the self-normalized partial sums from the NA random sequences have been obtained. However, since the denominator in the formula for the self-normalized partial sums contains random variables, the study of it theory for self-normalized partial sums of NA random variables is very difficult, and so far, there are very few research results in this field. Thus, this is a challenging, difficult and meaningful research topic. The purpose of this article is to establish the ASCLT for the self-normalized partial sums of NA random variables. We will show that the ASCLT holds under a fairly general growth condition on d, namely if d = exp(ln α ), 0 α < /2. Definition. Random variables X, X 2,..., X n, n 2, are said to be negatively associated (NA) if for every pair of disoint subsets A and A 2 of {, 2,..., n}, cov( f (X i ; i A ), f 2 (X ; A 2 )) 0, where f and f 2 are increasing for every variable (or decreasing for every variable) functions such that this covariance exists. A sequence of random variables {X i ; i } is said to be NA if its every finite subfamily is NA. The concept of negative association was introduced by Alam and Saxena [8] and Joag-Dev and Proschan [9]. Statistical test depends greatly on sampling. The random sampling without replacement from a finite population is NA, but is not independent. Due to the wide applications of NA sampling in multivariate statistical analysis and reliability theory, the it behaviors of NA random variables have received extensive attention recently. One can refer to: Joag-Dev and Proschan [9] for fundamental properties, Matuła [20] for the three series theorem, Su et al. [2] for the moment inequalities and wea convergence, Shao [22] for the Rosenthal type inequality and the Kolmogorov exponential inequality, Wu and Jiang [23] for the law of the iterated logarithm, Wu [24] for almost sure it theorems, Wu and Chen [25, 26] for strong representation results of Kaplan-Meier estimator and the Berry-Esseen type bound in ernel density estimation. In the following, a n b n denotes a n /b n = and the symbol c stands for a generic positive constant which may differ from one place to another. We assume that {X, X n } n N is a stationary sequence of NA random variables. By Newman [27], σ 2 := EX =2 EX X always exists and σ 2 [0, VarX]. Furthermore, if σ 2 > 0, then VarS n nσ 2. For each n, the symbol S n /V n denotes self-normalized partial sums, where: S n = n i= X i, V 2 n = n i= X2 i. For every i n, let: X ni := ni(x i < n) + X i I( X i n) + ni(x i > n), S n := X ni, V 2 n := X 2 ni, V 2 n, := X 2 ni I( X ni 0), V 2 n,2 := X 2 ni I( X ni < 0), i= i= i= i= σ 2 n := Var S n, δ 2 n := E X 2 n, δ2 n, := E X 2 n I( X n 0), δ 2 n,2 := E X 2 n I( X n < 0). Obviously, δ 2 n = δ 2 n, + δ2 n,2, E V n 2 = nδ 2 n = nδ 2 n, + nδ2 n,2. Our theorem is formulated in a more general setting.

3 Wu and Jiang / Filomat 3:5 (207), Theorem.. Let {X, X n } n N be a stationary sequence of NA random variables satisfying: EX = 0, 0 < EX 2 <, σ 2 > 0, EX 2 I(X 0) > 0, EX 2 I(X < 0) > 0, () and Then, σ 2 n β 2 E V 2 n = β 2 nδ 2 n for some constant β > 0. (2) Suppose that 0 α < /2 and set: d = exp(lnα ), = = d. = { } S d I x = Φ(x) a.s. for any x R. (4) βv By the terminology of summation procedures (see e.g. Chandraseharan and Minashisundaram [28], p. 35), we have the following corollary. Corollary.2. Theorem. remains valid if we replace the weight sequence {d } N by any {d } N such that 0 d d, = d =. Remar.3. If {X, X n } n N is a sequence of independent random variables then, (2) holds with β =. 2. Proofs The following four lemmas below play an important role in the proof of Theorem.. Lemma 2. is due to Joag-Dev and Proschan [9], Lemma 2.2 has been stated by Su et al. [2], Lemma 2.3 has been established by Wu [7], and Lemma 2.4 is of our authorship; due to its length, the proof of Lemma 2.4 is given in Appendix. Lemma 2.. (Joag-Dev and Proschan [9]) If {X i } i N is a sequence of NA random variables and { f i } i N is a sequence of nondecreasing (or nonincreasing) functions, then { f i (X i )} i N is also a sequence of NA random variables. Lemma 2.2. (Su et al. [2]) Let {X i } i N be a sequence of NA random variables with zero mean and such that E X i p <, i =, 2,... if p 2. Then, E S n p c p p/2 E X i p + EX 2 i, where c p > 0 only depends on p. i= Lemma 2.3. (Wu [7]) Let {ξ, ξ n } n N be a sequence of uniformly bounded random variables. If there exist constants c > 0 and δ > 0, such that i= (3) then Eξ ξ c ( ) δ, for <, d ξ = 0 = a.s., where d and are defined by (3).

4 Wu and Jiang / Filomat 3:5 (207), Lemma 2.4. Suppose that the assumptions of Theorem. hold. Then: S E S d I βδ = x = Φ(x) a.s. for any x R, (5) d f = V 2,l δ 2,l V 2 E f,l = 0 a.s., l =, 2, (6) δ 2,l where d and are defined by (3) and f is a bounded function with bounded continuous derivatives. Proof of Theorem.. For any given 0 < ε <, note that for x 0 and n, { } S x βv Hence, { S x, i X i, V 2 βv ( + ε)δ2 { V 2 > ( + ε)δ2 S x βδ ( + ε) { V 2 > ( + } ε)δ2 ( X i > ). i= } } { i Xi > } I ( ) S x βv S I x βδ + I ( V 2 > ( + ) ε)δ2 + I ( X i > ), for x 0. ( + ε) Similarly, we have for any given 0 < ε < and x < 0, ( ) S S I x I βv x βδ + I ( V 2 ( ) ε)δ2 + I ( X i > ). ( ε) Furthermore, we get ( ) S S I x I βv x βδ I ( V 2 ( ) ε)δ2 I ( X i > ), for x 0, ( ε) I ( ) S x βv S I x βδ I ( V 2 > ( + ) ε)δ2 I ( X i > ), for x < 0. ( + ε) Hence, in order to establish (4), it suffices to prove: i= i= i= i= S d I x ± ε βδ = Φ(x ± ε) a.s., (7) = = d I( V 2 > ( + ε)δ2 ) = 0 a.s., (8)

5 Wu and Jiang / Filomat 3:5 (207), = d I( V 2 ( ε)δ2 ) = 0 a.s., (9) d I ( X i > ) = 0 a.s., (0) = i= for any ε > 0. Firstly, we prove (7). By 0 = EX = EXI( X ) + EXI( X > ) and EX 2 <, we have EXI( X ) = EXI( X > ) and x x 2 P( X > x) = 0, and consequently E S EXI( X ) + 3/2 P( X > ) = EXI( X > ) + o( ) EX 2 I( X > ) + o( ) = o( ). This, and the fact that δ 2 EX2 <, when, imply S E S I βδ ± εx α I S ± εx βδ I S E S βδ ± εx + α for any α > 0. Thus, by (5), we have as n, Φ( ± εx α) S E S d I D ± εx α n = βδ S d I D ± εx n = βδ S E S d I D ± εx + α n βδ = Φ( ± εx + α) a.s. () Letting α 0 in (), we obtain that (7) holds. Now, we prove (8). Since E V 2 = δ2, V 2 = V 2, + V 2,2, E V 2,l = δ2,l, and δ2,l δ2, l =, 2, it follows that I( V 2 > ( + ε)δ2 ) = I( V 2 E V 2 > εδ2 ) I( V 2, E V 2, > εδ2 /2) + I( V 2,2 E V 2,2 > εδ2 /2) I( V 2, > ( + ε/2)δ2, ) + I( V 2,2 > ( + ε/2)δ2,2 ). Therefore, by the arbitrariness of ε > 0, in order to prove (8), it suffices to show that, for l =, 2, = d I( V 2,l > ( + ε)δ2,l ) = 0 a.s. (2)

6 Wu and Jiang / Filomat 3:5 (207), For a given ε > 0, let f denote a bounded function with bounded continuous derivatives, such that I(x > + ε) f (x) I(x > + ε/2). Obviously, X 2 ni I( X ni 0) is monotonic on X i, and thus, by Lemma 2., { X 2 ni I( X ni 0)} n,in is also a sequence of NA random variables. From Lemma 2.2, the Marov inequality and the facts that E V 2, = δ2, and δ 2, EX2 I(X 0) > 0, we get P ( V 2, > ( + ε/2)δ2, ) = P ( V 2, E V 2, > εδ2, /2) c E( V 2, E V 2, )2 c E X 4 I( X 0) 2 c EX4 I(0 X ) + 2 P( X > ). (3) Using EX 2 <, we have x 2 P( X > x) = o(), as x. Hence, EX 4 I(0 X ) = This and (3) imply 0 P ( X I(0 X ) > t ) 4t 3 dt c P( X > t)t 3 dt = o()tdt = o(). 0 0 P ( V 2, > ( + ε/2)δ2,) 0. Therefore, it follows from (6) and the Toeplitz lemma that 0 = d I ( V 2, > ( + ) ε)δ2, = d E f = V 2, δ 2, d f = V 2, δ 2, d EI ( V 2 D, > ( + ) ε/2)δ2, n = d P( V 2, > ( + ε/2)δ2, ) = 0 a.s. Hence, (2) holds for l =. Using similar methods to those used in the proof of (2) for l =, we can prove (2) for l = 2. Consequently, (8) holds. Moreover, applying identical methods to those used in the proof of (8), we can prove (9). Finally, we shall prove (0). Note that E I ( X i > ) i= P( X i > ) = P( X > ) 0, as. i= Therefore, I ( X i > ) 0 a.s., as. i= Thus, by the Toeplitz lemma,

7 Wu and Jiang / Filomat 3:5 (207), d I ( X i > ) 0 = i= a.s. Hence, (0) holds. This completes the proof of Theorem.. 3. Appendix As it has been mentioned, we give the proof of Lemma 2.4 in this part of our paper. Proof of Lemma 2.4. Obviously, X ni is monotonic on X i, and thus, by Lemma 2., { X ni } n,in is also a sequence of NA random variables. By the cental it theorem for NA random variables and the properties that: σ 2 n β 2 nδ 2 n, δ 2 n EX 2 > 0 as n (see conditions () and (2)), we get S n E S n βδ n n d N, where d denotes the convergence in distribution and N denotes the standard normal random variable. This implies that for any function (x), which is bounded and has bounded continuous derivatives ( ) S n E S n E E (N), as n, βδ n n Hence, by the Toeplitz lemma, we obtain S E S d E D n βδ = E (N). = On the other hand, it follows from Theorem 7. of Billingsley [29] and Section 2 of Peligrad and Shao [9] that (5) is equivalent to = S E S d βδ = E (N) a.s. Hence, in order to prove (5), it suffices to show that S E S d βδ E S E S = 0 a.s., (4) βδ = for any from the class of bounded functions having bounded continuous derivatives. Let for, S E S ξ = βδ E S E S βδ. Observe that, for any <, we get,

8 Wu and Jiang / Filomat 3:5 (207), Eξ ξ = Cov S E S S βδ, E S βδ = Cov S E S S βδ, E S βδ i=+ ( X i E X i ) βδ + Cov S E S βδ, i=+ ( X i E X i ) βδ =: I + I 2. Clearly, since is a bounded Lipschitz function, there exists a constant c > 0 such that (x) c, (x) (y) c x y, for any x, y R. As { X i },i is a sequence of NA random variables, as well as Lemma 2.2 and condition δ 2 n EX 2 < hold, we obtain that I c E i= ( X i E X i ) c E ( i= ( X i E X i ) ) 2 ( ) EX 2 /2 c c. It follows from the definition of negative association that S E S i=+, ( X i E X i ) are NA and the βδ βδ assumption that is a bounded function with bounded continuous derivatives. Thus, Lemma 2.3 of Zhang [30] is applied with: f (x) := I(x < ) + xi( x ) + I(x > ), (y) := I(y < ) + yi( y ) + I(y > ); the stationarity of {X i }, and the facts that EX 2 < and σ 2 > 0 imply m=2 Cov(X, X m ) <. Therefore, I 2 ccov = S E S, c Cov ( X l E X l ), c c l= l= i=+ Cov(X l, X i ) l+ l= m= l+2 i=+ ( X i E X i ) Cov(X, X m ) c Cov(X, X m ) c m=2 ( ) /2. Hence, by Lemma 2.3, (4) holds. δ 2, ( X i E X i ) i=+ Now, we prove (6). Let V 2 V, η = f 2 E f, for any <. δ 2,

9 Wu and Jiang / Filomat 3:5 (207), Since { X 2 ni I( X ni 0)} n,in is a sequence of NA random variables, we have δ 2, EX2 I(X 0) and 0 < EX 2 I(X 0) <. Thus, using Lemma 2.3 of Zhang [30] twice is used with: f (x) := x 2 I(0 x ) + I(x > ), (y) := y 2 I(0 y ) + I(y > ), we have for <, Eη η = = V Cov 2 V f, δ 2 2, f, δ 2,, V Cov 2 V f, δ 2 2 V 2, f, δ 2 f, X i= 2 i I( X i 0) δ 2,,, V + Cov 2 V f, δ 2 2, f, X i= 2 i I( X i 0) δ 2,, E ( X i= 2 i I( X i 0) ) V 2, V2 c ccov,, X i= 2 i I( X i 0) c c c c c + c ( l= i=+ Cov ( X 2 l I( X l 0), X 2 i I( X i 0) ) l= i=+ ) /2 Cov(X l, X i ) c ( ) /2. By Lemma 2.3, (6) holds. This completes the proof of Lemma 2.4. Acnowledgements The authors are very grateful to the referees and the Editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. References [] Brosamler, G.A., 988. An almost everywhere central it theorem. Mathematical Proceedings of the Cambridge Philosophical Society, 04, [2] Schatte, P., 988. On strong versions of the central it theorem. Mathematische Nachrichten, 37, [3] Lacey, M. T., Philipp, W., 990. A note on the almost sure central it theorem. Statistics and Probability Letters, 9, [4] Ibragimov, I.A., Lifshits, M., 998. On the convergence of generalized moments in almost sure central it theorem. Statistics and Probability Letters, 40, [5] Beres,I., Csái, E., 200. A universal result in almost sure central it theory. Stochastic Processes and their Applications, 94, [6] Hörmann, S., Critical behavior in almost sure central it theory. Journal of Theoretical Probability, 20, [7] Miao, Y., Central it theorem and almost sure central it theorem for the product of some partial sums. Proceedings of the Indian Academy of Sciences C. Mathematical Sciences, 8 (2), [8] Wu, Q.Y., 20. Almost sure it theorems for stable distribution. Statistics and Probability Letters, 8 (6): [9] Peligrad, M., Shao, Q.M., 995. A note on the almost sure central it theorem for wealy dependent random variables. Statistics and Probability Letters, 22,3-36.

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