The Inequality And Weak Convergence For Negatively Associated Random Variables

In this paper, we study some problems of limit theorems for negatively associated(NA) random variables.In the 1980's, the concept of negatively associated random variables , which include the concept of independent random variables, was introduced by Alam and Saxena(1981), Blok, Savits and Shaked(1982), Joag-Dev and Proschan(1983) and Joag-Dev, K.(1983). Definition : A finite family of random variables ,X₁, , X_n( n≥2)are said to be negatively associated (NA) if for,every pair of disjoint subsets A1 ,A₂ of{1 ,2, , n}, COV(f₁( X_i , i∈A₁),f₂( X_j ,j∈A₂))≤0, whenever f₁ and f₂ are real coordinatewise increasing and the covariance exists. An infinite family is negatively associated if every finite subfamily is negatively associated.Since the concept of NA random variables has a lot of application, multivariate statistical analysis, reliability theory, percolation theory, many engineering problems and risk analysis theory, various aspects of NA random variables are significant and have been investigated by a lot of scholars.This paper has three parts, the first chapter is about the central limit theorem for NA sequence. The second chapter is about the inequality for NA sequence. The third chapter is about the weak convergence for NA sequence. The chief results of first chapter are as follows. Theorem 1.1 Let { X j,j∈N} be a NA sequence with EX j=0, if the following conditions are satisfied:Theorem 1.2 Let {θn , n≥1} be a i.i.d. random variables,ψ= {φ( x ) :φ( x)} be a nonnegative And nondecraesing odd functions.φ( x )/ x ² and x ³/φ( x) be nondecreasing functions. If there existsφ∈ψsuch that Eφ(θi) is finite, then there exists positive constants K, then The second results of second chapter are as follows: Theorem 2.1 Let X 1 , X ²X n为be NA random variables,Then for any x>0,y>0.we have furthermoreTheorem 2.2 Let X₁ , X₂…X_n be NA random variables with E X_j=0, j=1,2, n, E |Z_j|^p<∞,for some p≥2.Let ,then for any t>p/2, x>0,we have Theorem2.3 Let { X_j, j∈N} be a NA sequence with zero mean. for some P≥2 Theorem 2.4 Let g ( x ):R→R⁺ be a continuous odd function which is increasing on (0, +∞),such that Eg ( X_j) <∞, j = 1,2, n. If 0 < An<∞, then for any t>0,we haveTheorem 2.8 Let p≥1,{ X_i ,1≤i≤n} w ithE X_ip<∞,1≤i≤n with mean zero random variables, and { X_i*,1≤i≤n} a sequence of independent random variables with X_i and X_i* have same distribution for each i=1,2, ,n, thenTheorem 2.9 Let { X 1,1≤i≤n} be a NA sequence, then for any x>0,a>0,0<α<1,we have The third results of second chapter are as follows:Theorem 3.1 Let { X_j,j∈N} be a stationary NA sequence with,m'nt,n∈N} Convergence in distribution to a Wiener processon C[0,T].Theorem 3. 2 {W_n(t), 0