NEGATIVELY DEPENDENT BOUNDED RANDOM VARIABLE PROBABILITY INEQUALITIES AND THE STRONG LAW OF LARGE NUMBERS

Let X 1 , … , X n be negatively dependent uniformly bounded random 
variables with d.f. F ( x ) . In this paper we obtain bounds for the probabilities P ( | ∑ i = 1 n X i | ≥ n t ) and \varepsilon } \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> P ( | ξ ˆ p n − ξ p | > ϵ ) where ξ ˆ p n is the 
sample p th quantile and ξ p is the p th quantile of F ( x ) . Moreover, we 
show that ξ ˆ p n is a strongly consistent estimator of ξ p under mild 
restrictions on F ( x ) in the neighborhood of ξ p . We also show that ξ ˆ p n 
converges completely to ξ p .


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 bivar- iate positive and negative dependence were introduced by Esary and Proschen [7].Also Esary, Proschen and Walkup [8] introduced a concept of association which im- plied a strong form of positive dependence.Their concept has been very useful in reli- ability 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- ables.
Definition 1" if 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 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.
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, P[IXl >1 <c. (5) n--1 In the following example, we will show that the ND properties are not preserved for absolute values and squares of random variables.
Example: Let (X, Y) have the following p.d.f: 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).
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 Lemma3: Let X be a r.v.such that E(X)-O and XI <_c<oc a.e.
every real number h we have Then for Ee hX < eh2c2.
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.
Then for any t > O and c b-a P[S n >_ nt] <_ exp c2.j.
k=l Proof: By Theorem 2 and Corollary 1, for each > 0 we have Hence, for a-1 we obtain the strong law of large numbers for negatively depen- dent 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 suffi- ciently large n.
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].