Concepts of negative dependence have been useful in developing laws of large numbers (cf: Taylor, Patterson and Bozorgnia [1]). Chung-type laws of large numbers for arrays of independent random variables were developed by Taylor, Patterson and Bozorgnia in [2].

Definition 1.1 Two random variables X and Y are pairwise negatively dependent (ND) if

for all.

Let (W, F, P) denote a probability space.

Definition 1.2. The sequence of random variables is said to be conditionally negatively dependent if there exists a sub s-field z of F such that for each positive integer m

where denotes the conditional probability of the random variable X being in the Boral set given the sub-s field z. Negatively dependent random variable are conditionally negatively dependent with respect to the trivial s-field.

Throughout this paper will denote rowwise conditionally independent random variables such that for all n and k. The major result of this paper shows that

where complete convergence is defined (Hsu and Robbins [3]) by

Here is a function on a separable Banach space toR. In the next section of this paper, strong laws of large numbers for arrays of rowwise conditionally negatively dependent random variables.

In this section, several lemmas are used in the proof of the major result. The first lemma will be presented without proof.

Lemma 2.1. Let X and Y be pairwise negatively dependent random variables. Then

Lemma 2.2. Let X and Y be pairwise negatively dependent random variables. Then

Proof: For X and Y negatively dependent, we have by Lemma 2.1

Theorem 2.1 Let be an array of rowwise conditionally negatively dependent random variables. If

a) (5)

and for all h > 0

b) (6)

where is the conditional expectation with respect to an appropriate s-field that gives conditional negative dependence. Then

.

Proof. Let h > 0 be given. By Markov’ inequality

By Lemma 2.2, the first term in Equation (7) is bounded by

.

The second term of Equation (7) is finite by Equation (6). Thus, the result follows.

Theorem 1.2 can be extended to R^m. The next definition is a crucial type p inequality used to define a form of negative dependence (cf. Patterson, Taylor, and Bozorgnia [4]).

Definition 3.1. Random elements, in a type p Banach space are said to be type p negatively dependent if and if there exist a finite positive constant C such that

for all n ≥ 1.

Coordinatewise (with respect to the standard basis) negative dependence in R^m can yield type 2 negative dependence. To see this for rowwise random elements, let be random elements in R^m such that for 1≤ i ≤ m, n, k ≥ 1. Then

Theorem 3.1 Let be an array of rowwise conditionally coordinatewise negatively random elements in R^m. If a)(10)

and for all h > 0 b)(11)

where is the conditional expectation with respect to an appropriate s-field that gives conditional negative dependence, then

Proof. The proof is similar to that of Theorem 2.1.