THE STRONG LAW OF LARGE NUMBERS

Applied Mathematics and Stochastic Analysis, 13:3 (2000), 261-267. NEGATIVELY DEPENDENT BOUNDED RANDOM VARIABLE PROBABILITY INEQUALITIES AND THE STRONG LAW OF LARGE NUMBERS M. AMINI and A. BOZORGNIA Ferdowsi University Faculty of Mathematical Sciences E-maih Mashhad, Iran Bozorg@science2.um.ac.ir (Received January, 1998; Revised January, 2000) Let Xl,...,X n be negatively dependent uniformly bounded random variables with d.f. F(x). In this paperwe obtain bounds for the^ probabilities P(I Y=IXil >_nt) and P(l(pn-pl >e) where pn is the sample pth ^quantile and p is the pth quantile of F(x). Moreover, we show that pn is a strongly consistent estimator of p under mild^ restrictions on F(x) in the neighborhood of p. We also show that (pn converges completely to p. Key words: Probability Inequalities, Strong Law of Large Numbers, Complete Convergence. AMS subject classifications: 60E15, 60F15. 1. Introduction In many stochastic models, the assumption that random variables are independent is not plausible. Increases in some random variables are often related to decreases in other random variables so an assumption of negative dependence is more appropriate than an assumption of independence. Lehmann [12] investigated various conceptions of positive and negative dependence in the bivariate case. Strong definitions of bivariate positive and negative dependence were introduced by Esary and Proschen [7]. Also Esary, Proschen and Walkup [8] introduced a concept of association which implied a strong form of positive dependence. Their concept has been very useful in reliability theory and applications. Multivariate generalizations of conceptions of dependence were initiated by Harris [9] and Brindley and Thompson [4]. These were later developed by Ebrahimi and Ghosh [6], Karlin [11], Block and Ting [2], and Block, Savits and Shaked [1]. Furthermore, Matula [13] studied the almost sure convergence of sums of negatively dependent (ND) random variables and Bozorgnia, Patterson and Taylor [3] studied limit theorems for dependent random variables. In this paper we study the asymptotic behavior of quantiles for negatively dependent random vari- Printed in the U.S.A. @2000 by North Atlantic Science Publishing Company 261

262 M. AMINI and A. BOZORGNIA ables. Definition 1" if H= IIY= The random variables X1,...,X n are pairwise negatively dependent P(Xi <_ xi, X j <_ xj) <_ P(X <_ xi)p(x j <_ xj) for all x i, x j (1) e R, 7 j. It can be shown that (1) is equivalent to, P(X > xi, X j > xj) <_ p(x > xi)p(x j > xj) for all xj, x E # _ j. Defmition 2: The random variables X1,... X n - are said to be ND if we have P(NY= I(Xj <- xj)) 1P(XJ xj), (3) and P(N= I(Xj > xj)) 1P(Xj > xj), (4) for all Xl,...,xn R. Conditions (3) and (4) are equivalent for n- 2. However, Ebrahimi and Ghosh [6] show that these definitions do not agree for n >_ 3. An infinite sequence {Xn, n >_ 1} is said to be ND if every finite subset {X1,...,Xn} is ND. Definition 3: For parametric function g(0), a sequence of estimators {Tn, n >_ 1} is strongly consistent if Tn--og(O a.e. Definition 4: The sequence {Xn, n > 1} of random variables converges to zero completely (denoted limn,ocx n 0 completely) if for every > 0, In the following example, we will for absolute values and squares of random variables. Example: Let (X, Y) have the following p.d.f: P[IXl >1 <c. (5) n--1 show that the ND properties are not preserved f(- 1, 1) f(1,0) 0, f(- 1,0) f(0,0) f(0, 1) f(0,1) f(1,1) 1/9, f(-1,1)- f(1,-1)-2/9. Then X and Y are ND random variables since for each x, y G R we have (ii) F(x,y) <_ Fx(x)Fy(y ). X and V-y2 are not ND random variables because for 0<v<l we have -1 <x<0 and

Negatively Dependent Bounded Random Variable Probability Inequalities 263 F(x, v)- 1/9 > Fx(x)Fv(v -(3/9)(2/9). (iii) U X 2 and V y2 are not ND random variables nor are IX land YI since0_<u<l, 0<v<l we have F(u, v)- 1/9 > Fu(u)Vv(v -(2/9)(3/9). The following lemmas are used to obtain the main result in the next section. Detailed proofs of these lemmas can be founded in the Bozorgnia, Patterson and Taylor [3]. Lemma 1: Let {Xn, n > 1} be a sequence of ND random variables let {fn, n > 1} be a sequence of Borel functions all of which are monotone increasing (or all are monotone decreasing). Then {fn(xn),n > 1} is a sequence of NO random variables. Lemma 2: Let X1,...,X n be ND random variables and let tl,...,t n be all nonnegative (or all nonpositive). Then E[e i=1 < EetiXi i=1 Lemma3: Let X be a r.v. such that E(X)-O and XI <_c<oc a.e. every real number h we have Proof: For c- 1, see Chow [5]. replaced by X/c. Ee hx < eh2c2. Then for For general c, apply the c-1 result with X 2. An Extension of the Theorem of Hoeffding for ND Random Variables In this section we extended the theorem of Hoeffding (Theorem 1 below) and then obtain the strong law of large numbers for ND uniformly bounded random variables. Theorem 1: (Hoeffding [10]).Let X1,...,X n be independent random = variables satisfying P[a <_ X <_ b]- 1 for each where a < b, and let S n l(xk- EXk). Then for any t > O and c b-a P[S n >_ nt] <_ exp c2. j. Theorem 2: Let X1,...,X n be ND random variables satisfying P[a <_ X <_ b]- 1 for each where a<b, and let Sn- =I(Xk-EXk). Then for any t>o and c-b-a, Proofi We define Yk- Xk-EXk p[s _> ] _< txp 4: j. for k- 1,...,n. Then we have EY YI c a.e. Hence, k -0 and by Lemmas 2 and 3, we have

264 M. AMINI and A. BOZORGNIA p(sn >_ nt) <_ exp(- nth)ee hsn _< exp[- nth] H EehYk <- exp[- nth + nh2c2]. k-1 The right side of this inequality attains its minimum value exp[- nt2] for h- t._ 4c 2 2c 2" Thus, for each t > 0, P[Sn>_nt]<_exp 4c2 j. V! Corollary 1: Under the assumptions of Theorem 1, for every t > 0 P[IS.I >-nt]<-2exp 4c2 ]. P( Sn > nt) P[S n >_ nt] + P[ S n >_ nt] P[S n >_ nt] + P[S n >_ nt] <_ 2exp nt2] 4c 2 j wheres2n- E=lZkandZk- -Yk, k-1,...,n. Theorem 3: Under the assumptions of Theorem 2, for every a > 1/2 we have 1 n n --a Z (Xk- EXk)-*O completely. k=l Proof: By Theorem 2 and Corollary 1, for each > 0 we have ZP(ISnl >na)<2exp n--1 n--1 2n2a l j < 4c 2 ] Hence, for a- 1 we obtain the strong law of large numbers for negatively dependent uniformly bounded random variables. 3. Asymptotic Behavior of Quantiles for ND Random Variables The following two theorems and one corollary given conditions under which pn is contained in a suitably small neighborhood of p with probability one for all sufficiently large n. Theorem 4: Let Xl,...,X n be ND random variables with d.f. F(x). Let O < p < 1. Suppose that p is the unique solution x of F(x- <_ p <_ F(x). Then for every > 0 and n we have P( "p.- p > ) <_ 2exp(- n62/4) (6)

Negatively Dependent Bounded Random Variable Probability Inequalities 265 where min{f(p + ) p, p F(p )} and pn is the sample pth quantile. Proof: We define For every^ > 0 we have P[ pn- p > ] P[pn > + p] + P[pn < p- ] Pip > Fn(p + )] + PIp < Fn( p V I[x > p+ e] and U I[x < p_e] for 1,2,...,n. Since X,...,X n are ND random variables, by Lemma 1 ND random variables. Hence, by Theorem 2 23 have U,...,U n and V1,..., V n are P(P > Fn(p + )) P E (Yi- E(Yi) > nl) -< exp i=1 where 1 F(p + )- p. Similarly we have P[p < Fn(v-)] P (Ui-EUi) > n52 exp.. 4... where 5 p- F(p- ). Define 5e min{51, 52}. Thus we have (6). H Corolly 2: Let X1,...,X n be ND random variables with d.f. F(x) and let pn be the sample pth quantile. Then pn----p completely as n--,cx. Proof: By Theorem 4 we have n1( ) P[l p -p[ >1< 4 n=l Hence pn---p completely. n 2 exp n52 <" By the Borel-Cantelli lemma, we have pn---p a.e. Thus pn is a strongly consistent estimator of (p. Theorem 5: Let X1,...,X n be ND random variables with d.f. F(X). Let 0 < p < 1. Suppose that F is differentiable at p with F (p) some 1 fl > O and O < a <_- f(p) > O. Then for (2 + fl)lna(n) Vl Proof: Since F is continuous at p with F (p)> 0, p is a unique solution of F(x-) < p < F(x) and F(p)- p. Thus, we may apply Theorem 4. Writing (2 + 3)lnC(n) we have (2 /3)inC (n) r(p + n) P nf((p) + (n) >

266 M. AMINI and A. BOZORGNIA where n is sufficiently large. Thus for all sufficiently large n, where Similarly p F(p n) nf(p) + (n) > (2 + fl)21n(n) n(2 /4 > (2 + )21n2a(n) (2 + )21n(n) n 4n2a- 1 4 6% min{f(p + en) P, P F(p en)}" Hence, for a constant c we have n--1 n=l n (+/2) 2 which completes the proof. Let X1,...,X n be independent random variables with d.f. F(x). Then in this case, all the above theorems and corollaries are true. In particular, Theorems 4 and 5 are extensions of Theorem 2.3.1 and Lemma B, respectively, pages 96 of Settling [14]. Acknowledgement We wish to express our appreciation and gratitude to the referees for their suggestions which improved the exposition. References [1] [2] [3] [4] [5] [8] [9] Block, H.W., Savits, T.H. and Shaked, M., Some concepts of negative dependence, Ann. Probab. V10:3 (1982), 7???-772. Block, H.W. and Ting, M.L., Some concepts of multivariate dependence, Comm. Star. Theor. Math. A10:8 (1981), 762. Bozorgnia, A., Patterson, R.F. and Taylor, R.L., Limit theorems for dependent random variables, Proceedings of the WCNA 92, Tampa, Florida (1996), 1639-1650. Brindley, E. and Thompson, W., Dependent and aging aspects of multivariate survival, J. Amer. Star. Assoc. 67 (1972), 822-830. Chow, Y.S., Some convergence theorems for independent random variables, Ann. Mat. Statist. 37 (1966), 1482-1493. Ebrahii, N. and Ghosh, M., Commun. Star. Theory. Math. 4 (1981), 307-337. Esary, J.D. and Proschan, F., Relationships among some concepts of bivariate dependence, Ann. Math. Star. 43 (1972), 651-655. Esary, J.D., Proschan, F. and Walkup, D.W., Association of random variables with applications, Ann. Math. Star. 38 (1967), 1466-1474. Harris, R., A multivariate definition for increasing hazard rate distributions, Ann. Math. Star. 41 (1970), 713-717.

Negatively Dependent Bounded Random Variable Probability Inequalities 267 [lo] [12] [14] Hoeffding, W., Probability inequalities for sums of bounded random variables, J. Amer. Slat. Assoc. 58 (1963), 13-30. Karlin, S., Classes of orderings of measures and related correlation inequalities, J. Mull. Analysis 10 (1980), 499-516. Lehmann, E., Some concepts of dependence, Ann. Math. Statist. 37 (1966), 1137-1153. Matula, P., A note on the almost sure convergence of sums of negatively dependent random variables, Slat. Probab. Letters 15 (1992), 209-213. Settling, R.J., Approzimation Theorems of Mathematical Statistics, John Wiley and Sons, New York 1980.

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