In this paper, we obtain some Rosenthal’s type inequalities for NEGATIVELY orthant DEPENDENT (NOD) RANDOM VARIABLES.

Let { X n≥  1} be a stationary sequence of pair wise negative quadrant DEPENDENT RANDOM VARIABLES with survival function  F( x)=P[X >x]. The empirical survival function Fn(x) based on X1, X2,… X n is proposed as an estimator for Fn ( x). Strong consistency and point wise as well as uniform of Fn (x) are discussed.    

We discuss in this paper the strong convergence for weighted sums of NEGATIVELY orthant DEPENDENT (NOD) RANDOM VARIABLES by generalized Gaussian techniques. As a corollary, a Cesaro law of large numbers of i.i.d. RANDOM VARIABLES is extended in NOD setting by generalized Gaussian techniques.  

In this paper, we obtain the upper exponential bounds for the tail probabilities of the quadratic forms for NEGATIVELY DEPENDENT subgaussian RANDOM VARIABLES. In particular the law of iterated logarithm for quadratic forms of inDEPENDENT subgaussian RANDOM VARIABLES is generalized to the case of NEGATIVELY DEPENDENT subgaussian RANDOM VARIABLES.

In this paper, strong laws of large numbers (SLLN) are obtained for the sums Σn i=1Xi , under certain conditions, where {Xn ,n≥1} n is a sequence of pairwise NEGATIVELY DEPENDENT RANDOM VARIABLES.

In this paper, we generalize a theorem of Shao [12] by assuming that {Xn} is a sequence of linear NEGATIVELY DEPENDENT RANDOM VARIABLES. Also, we extend some theorems of Chao [6] and Thrum [14]. It is shown by an elementary method that for linear NEGATIVELY DEPENDENT identically RANDOM VARIABLES with finite p –th absolute moment (p³2) p the weighted sums 1/An åni=1 aniXi converge to zero as where An=n1/p and (åni=1 a2ni) {an,i} a is an array of real numbers. Moreover, we prove the almost sure convergence for weighted sums åni=1 aniX, n³1, when {Xi, i ³ 1} i is a sequence of pairwise negative quadrant dependence stochastically bounded RANDOM VARIABLES under some suitable conditions on an,i.

 In the following work we present a proof for the strong law of large numbers for pairwise NEGATIVELY DEPENDENT RANDOM VARIABLES which relaxes the usual assumption of pairwise independence. Let {Xij} be a double sequence of pairwise NEGATIVELY DEPENDENT RANDOM VARIABLES. If p {│Xij│≥t}≤ p{│X│≥t} for all non-negative real numbers t and E│X│p log+ │X│<∞, for 1 < p < 2, then we prove that  Σ i-1 m  Σ j-1 n (Xij-EXij)/ (mn)1/p → 0 a.e  as m,n  →∞ . In addition, it also converges to 0 in L1. The results can be generalized to an r-dimensional array of RANDOM VARIABLES under condition E │X│p (log+ │X│r-1  < ∞, thus, extending Choi and Sung’s result [7] of one dimensional case for NEGATIVELY DEPENDENT RANDOM VARIABLES.  

In this paper, first we obtain some inequalities for the maximum of partial sums ∑nk=1Xn where{Xn ,n≥1} is a sequence of NEGATIVELY DEPENDENT (ND) RANDOM VARIABLES. Then we apply these inequalities to generalized some classical strong limit theorems in probability, if E[Xn │F­n-1]=0, and Fn  =σ(X1 ,X2 ,…Xn), for every n≥1.

In this paper, we will discuss the concept of almost sure convergence for specific groups of fuzzy RANDOM VARIABLES. For this purpose, we use the type of generalized Chebyshev inequalities. Moreover, we show the concept of almost sure convergence of weighted average pairwise NQD of fuzzy RANDOM VARIABLES.

In this paper, we extend and generalize some recent results on the strong laws of large numbers (SLLN) for pairwise inDEPENDENT RANDOM VARIABLES [3]. No assumption is made concerning the existence of independence among the RANDOM VARIABLES (henceforth r.v.’s). Also Chandra’s result on Cesàro uniformly integrable r.v.’s is extended.