EXPONENTIAL PROBABILITY INEQUALITY FOR LINEARLY NEGATIVE QUADRANT DEPENDENT RANDOM VARIABLES

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1 Commun. Korean Math. Soc. (007), No. 1, pp EXPONENTIAL PROBABILITY INEQUALITY FOR LINEARLY NEGATIVE QUADRANT DEPENDENT RANDOM VARIABLES Mi-Hwa Ko 1, Yong-Kab Choi, and Yue-Soon Choi Reprinted from the Communications of the Korean Mathematical Society Vol., No. 1, January 007 c 007 The Korean Mathematical Society

2 Commun. Korean Math. Soc. (007), No. 1, pp EXPONENTIAL PROBABILITY INEQUALITY FOR LINEARLY NEGATIVE QUADRANT DEPENDENT RANDOM VARIABLES Mi-Hwa Ko 1, Yong-Kab Choi, and Yue-Soon Choi Abstract. In this paper, a Berstein-Hoeffding type inequality is established for linearly negative quadrant dependent random variables. A condition is given for almost sure convergence and the associated rate of convergence is specified. 1. Introduction Lehmann [6] introduced a simple and natural definition of negative dependence: A sequence {X i, 1 i n} of random variables is said to be pairwise negative quadrant dependent (pairwise NQD) if for any real x i, x j and i j, P (X i > x i, X j > x j ) P (X i > x i )P (X j > x j ). Much stronger concept than NQD was considered by Joag-Dev and Proschan [4]: A sequence {X i, 1 i n} is said to be negatively associated(na) if for any disjoint subsets, A, B {1,,..., n} and any real coordinatewise increasing functions f on R A and g on R B, Cov(f(X i, i A), g(x i, i B)) 0. Instead of negative association, Newman [8] noticed that his method of proof yielding the central limit theorem for negatively associated sequence requires only that positive linear combinations of the random variables are NQD, i.e., the random variables are linearly negative quadrant dependent (LNQD). This notion of negative dependence was formulated by Newman [8] as follows: {X n, n N} is a sequence of LNQD random variables if for any disjoint subsets A, B of N and positive r i, the random vector ( i A r ix i, i B r ix i ) is NQD. Negatively associated sequence are LNQD and LNQD sequences are not necessarily NA, as it can be seen from examples in Newman [8] or Joag-Dev [3]. Received September, Mathematics Subject Classification. 60F15. Key words and phrases. exponential inequality, negatively associated, linearly negative quadrant dependent, almost sure convergence. 1 This work was supported by the SRC/ERC program of MOST/KOSEF(R ). Supported by KOSEF-R c 007 The Korean Mathematical Society

3 138 MI-HWA KO, YONG-KAB CHOI, AND YUE-SOON CHOI We note also that negative association and its weaker concepts are of considerable use in probability and statistics (cf. Joag-Dev and Proschan [4]; Newman [8] and the references there in). Newmann [8] was first to establish a central limit theorem for LNQD random variables, Zhang (000) proved a functional central limit theorem for LNQD random fields and Kim, Ko and Ryu (004) derived a general central limit theorem for weighted sum of LNQD random variables. Let X 1, X,... be random variables defined on the underlying probability space(ω, A, P) and set S n for sum of the first n random variables, n X i, and S n for S n /n. The problem of providing exponential bounds for the probabilities P ( S n ɛ)(ɛ > 0) is of paramount importance, both in probability and statistics. From a statistical viewpoint, such inequalities can be used, among other things, for the purpose of providing rates of convergence (both in the probability sense and almost surely) for estimates of various quantities. Especially so in a nonparametric setting, where the advantages of structure are not available to the investigator. Exponential inequalities for various kinds of random variables were studied extensively. Some of these exponential inequalities were studied by Devroye [1], Roussas [9] and Shao [10]. In this paper, a Bernstein-Hoeffding-type inequality is established for linearly negative quadrant dependent random variables. A condition is also given for almost sure convergence, and the associated rate of convergence is considered.. Main results For the formulation of the result to be made in this paper, the introduction of some notation is required. This notation is closely related to the way the proofs are carried out. Namely, for positive integers 1 p = p(n) < n and p, divide the set {1,,..., n} into successive groups each containing p elements. Let r = r(n) be the largest integer with: (.1) 0 < r < n, r, and pr n, which implies that n/pr 1. Thus, the set {1,,..., n} is split into r groups, each consisting of p elements; the remaining n pr < p elements constitute a set which may be empty. The basic assumption under which the main result in this paper is obtained is that the random variables X i s are mean zero LNQD and bounded, i.e., X i M, i 1. Define S n and ɛ n by (.) S n = 1 n n ( αm X i, ɛ n = ) 1 ( log n r ) 1,

4 EXPONENTIAL PROBABILITY INEQUALITY FOR LNQD RVS 139 where M is as in assumption and r is as in (.1), and α is an arbitrary constant > 1 (see discussion just prior to relation (3.1)). Then the exponential inequality to be established is the following: Theorem.1. Let {X i, i 1} be a sequence of LNQD random variables which are bounded, i.e., X i M and EX i = 0, and let S n and ɛ n be defined by (.). Then, (.3) P ( S n ɛ n ) 4 exp( crɛ n ), c = 9M for all sufficiently large n, n n 0. Furthermore, S n 0 a.s. at the rate 1/ɛ n. Finally we introduce an extension of Theorem in Hoeffding [] to the LNQD case. Theorem.. Let {X i, i 1} be a sequence of LNQD random variables such that a i X b i and EX i = 0 for all i 1. Then, for every t > 0, (.4) P ( S n t) exp[ n t /(b i a i ) ]. 3. Proof of the main results Lemma 3.1. Let {X i, 1 i n} be a sequence of LNQD random variables. Then (3.1) E(Π n exp(x i )) Π n E(exp X i ). Proof. Since {X i, 1 i n} is LNQD, X i and X i+1 + +X n, i = 1,,..., n 1, are NQD. Note that exp(x i ) and exp(x i X n ) are also NQD by Lemma of Matula [7]. Hence, E(Π n exp(x i )) = E[exp(X 1 ) exp(x + + X n )] E(exp X 1 )E(Π n i= exp(x i )) Π n (E exp(x i )) by induction. Set S n = n X i and S n = S n /n. With p and r as in the previous section, define the random variables U i, V i, i = 1,..., r and W n by (3.) U i = X (i 1)p X (i 1)p, (3.3) V i = X (i 1)p X ip, (3.4) W n = X pr X n, and (3.5) U n = 1 n r U i, V n = 1 n r V i, W n = W n n,

5 140 MI-HWA KO, YONG-KAB CHOI, AND YUE-SOON CHOI so that (3.6) S n = U n + V n + W n. We may now formulate the next lemma. Lemma 3.. Let U n be defined by (3.5), let ɛ n > 0, and suppose X i s are LNQD random variables with EX i = 0 and X i M for all i 1. Then ( ) rɛn P (U n ɛ n ) exp. Proof. Note that the random variables U 1,..., U r are LNQD and U i pm for all i. For some λ > 0 we have M (3.7) Ee λu n Π r (Ee (λ/n)ui ) by Lemma 3.1. At this point, we apply Lemma 1 in Devroye [1], which states that, if EX = 0 and a x b, then for every λ > 0, E exp(λx) exp[λ (b a) /8]. Take X = U i, so that U i pm and b a = pm. Then we obtain and hence E exp((λ/n)u i ) exp(λ p M /n ), (3.8) Π r Ee (λ/n)u i e λ M p r/n e λ M /8r, because λ M p r n = λ M ( ) pr λ M by (.1). 8r n 8r From (3.7) and (3.8) it follows that (3.9) Ee λu n e λ M /8r. Therefore, for ɛ n > 0 (3.10) P (U n ɛ n ) e λɛn+(λ M /8r). Minimizing, with respect to λ, the right-hand side in (3.10), we obtain (3.11) P (U n ɛ n ) e rɛ n /M for λ 0 = 4rɛ n /M. This completes the proof of the lemma. For the almost sure convergence purpose, we wish to have rɛ n /M = log n α (for any arbitrary α > 1), or equivalently αm log n (3.1) ɛ n =. r For this choice of ɛ n, λ 0 becomes 8α 1 (3.13) λ 0 = (r log n) M. Summarizing these observations, we have

6 EXPONENTIAL PROBABILITY INEQUALITY FOR LNQD RVS 141 Lemma 3.3. Under assumptions of Theorem.1 and with ɛ n specified by (3.1), it holds P (U n ɛ n ) n α. Remark 3.1. It is obvious that V n, as defined in (3.5) satisfies the same inequalities as U n in Lemmas 3. and 3.3. The following observation is meant to explain that we may dispense with W n as defined in (3.5). Lemma 3.4. Under assumptions of Theorem.1 and with ɛ n defined by (3.1), P (W n ɛ n ) = 0 for all sufficiently large n. Proof. W n consists of n pr terms and n pr < p. Then W n pm/n, so that P ( W n ɛ n ) P (M > nɛ n /p). The last expression, however, is 0, for all sufficiently large n, because αm nɛ n p = ( n log n p r this is so because n/pr 1. ) 1 ( ) = (αm ) 1 n (r log n) 1 ; pr Proof of Theorem.1. It consists, essentially, in combining Lemma 3., Remark 3.1 and Lemma 3.4. The random variables X i, i = 1,..., n, have the same properties as the random variables X i, i = 1,..., n. Thus, always P ( U n ɛ n ) = P (U n ɛ n ) + P ( U n ɛ n ) exp( rɛ n /M ), and similarly for P ( V n ɛ n ). Therefore, P ( S n 3ɛ n ) P ( U n ɛ n ) + P ( V n ɛ n ) + P ( W n ɛ n ) P ( U n ɛ n ) + P ( V n ɛ n ) (for every n n 0, say) Replacing ɛ n by ɛ n 3, we obtain, finally, 4 exp( rɛ n /M ). P ( S n ɛ n ) 4 exp( crɛ n ), c = 9M, n n 0. Remark 3.. The specification of ɛ n by (3.1), leads to the convergence S n 0 a.s. at the rate of 1/ɛ n. Regarding the latter part of the theorem, proceed as follows. For the value of ɛ n specified in (3.1), the rate of convergence is given by 1 r (3.14) =. ɛ n αm log n

7 14 MI-HWA KO, YONG-KAB CHOI, AND YUE-SOON CHOI Proof of Theorem.. Note that (3.15) E exp[λx i ] exp[λ (b i a i ) /8] by Lemma 3.1 in Devroye [1] and (3.16) P (S n t) exp( λnt)e n exp(λx i ) by Lemma 3.1. By (3.1), (3.15) and (3.16) we get (3.17) P (S n t) exp( λnt)e exp( λnt) [ λ exp 8 n exp(λx i ) n E exp(λx i ) n ] (b i a i ) λnt. By minimizing (with respect to λ) the right hand side in (3.17), the desired result (.4) follows. References [1] L. Devroye, Exponential inequalities in nonparametric estimation. In: Roussas, G. (Ed.) Nonparametric Functional Estimation and Related Topics, Kluwer Academic Publishers, Dordrecht (1991), [] W. Hoeffing, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), [3] K. Joag-Dev, Independence via un correlatedness under certain dependence structures, Ann. Probab. 7 (1983), [4] K. Joag-Dev and F. Proschan, Negative association of random variables with applications, Ann. Statist. 11 (1983), [5] T. S. Kim, D. H. Ryu, and M. H. Ko, A central limit theorem for general weighted sums of LNQD random variables and its applications, Rocky Mountain J. of Math., (accepted). [6] E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist. 37 (1966), [7] P. Matula, A note on the almost sure convergence of sums of negatively dependent random variables, Statist. Prob. Lett. 15 (199), [8] C. M. Newman, Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong, Y.L. (Ed.), Inequalities in Statistics and Probability, IMS Lectures Notes-Monograph Series, Hayward CA 5 (1984), [9] G. G. Roussas, Exponential probability inequalities with some applications. In: Ferguson, T. S., Shapely, L. S., MacQueen, J. B. (Ed.), Statistics, Probability and Game Theory, IMS Lecture Notes-Monograph Series, Hayward, CA 30 (1996), [10] Q. M. Shao, A comparison theorem on moment inequalities between negatively associated and independent random variables, J. Theo. Prob. 13 (000),

8 EXPONENTIAL PROBABILITY INEQUALITY FOR LNQD RVS 143 Mi-Hwa Ko Statistical Research Center for Complex Systems Seoul National University Seoul , Korea address: Yong-Kab Choi Division of Mathematics and Information Statistics Gyeongsang National University Kyungnam , Korea address: Yue-Soon Choi Division of Electric-Electronic-Information Engineering WonKwang University Jeonbuk , Korea address: