We propose a wavelet based stochastic regression function estimator for the estimation of the regression function for a sequence of PAIRWISE NEGATIVE QUADRANT DEPENDENT RANDOM VARIABLES with a common one-dimensional probability density function. Some asymptotic properties of the proposed estimator are investigated. It is found that the estimators have similar properties to their counterparts studied earlier in literature.    

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.    

In this paper, we discuss strong laws for weighted sums of PAIRWISE NEGATIVEly DEPENDENT RANDOM VARIABLES. The results on i.i.d case of Soo Hak Sung [9] are generalized and extended.

Let {Xj ; j³1} be a sequence of weakly NEGATIVE DEPENDENT (denoted by, WND) RANDOM VARIABLES with common distribution function F and let {qj ; j³1} be other sequence of positive RANDOM VARIABLES inDEPENDENT of {Xj ; j³1} and P[a£qj £b]=1 for some 0<a£b<¥ and for all j³1. In this paper, we study the asymptotic behavior of the tail probabilities of the maximum, weighted sums, RANDOMly weighted sums and RANDOMly indexed weighted sums of heavy-tailed weakly NEGATIVE DEPENDENT RANDOM VARIABLES, say, max1£j£nXj, Snj=1 cjXj , Snj=1 qjXj , and SNj=1 qjXj , respectively, where {cj ; 1£j£n} are n bounded positive real numbers and N is a nonNEGATIVE integer-valued RANDOM VARIABLES, inDEPENDENT of qi and Xi for all i³1. In fact, for a large class of heavy-tailed distribution functions, we show that the asymptotic relations,P[max1£j£n qjXj>x]~P[Snj=1 qjXj>x]~Snj=1 P[qjXj>x],hold as x®¥. Finally, if E(N)<¥ and also {qj ; j³1} is a sequence of identical inDEPENDENT positive RANDOM VARIABLES, then we prove that P[SNj=1 qjXj>x]~E(N). P[q1X1>x], as x®¥.

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 obtain some Rosenthal’s type inequalities for NEGATIVEly orthant DEPENDENT (NOD) RANDOM VARIABLES.

In this paper, we extend some moment inequalities for partial sums of NEGATIVE dependence (ND) RANDOM VARIABLES. Based on these inequalities, some useful inequalities are obtained.

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.

We consider an irreducible and aperiodic Markov chain {kn}n=0 over the finite state space E = {1, . . . , p} with positive regular transition matrix P = {pij} and additive component {Un} such that {Sn} = {(kn , Un)} is also a Markov chain over the state space e1 = E × R We prove a central and a local limit theorem for this chain when the probability density functions of {Sn}, conditional on the first and the last states of {kn}n=0, exist.

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.