Some Limit Theorems For Dependent Random Variables

This thesis is finished under the guidance of my tutor, professor Lin Zhengyan, during my master of science. It consists of three chapters:Chapter 1 Complete convergence for arrays of rowwise negatively associated random variablesHsu and Robbins introduced the concept of complete convergence in 1947. From then on, many authors devote the study to the complete convergence for independent random variables. In 1998, Hu et al. presented a theorem concerning the complete convergence for an array of rowwise independent random variables. Then, some authors discussed the proof and the assumptions of the former theorem, and gave some relatively results. In this chapter, we extend some results to the rowwise negatively associated random variables:Theorem 0.1.1 Let {X_nk,1≤k≤k_n,n≥1} be an array of rowwise negatively associated random variables, and {c_n, n≥1} a sequence of positive constants suchthat Σ_n=1^∞c_n=∞.Suppose that for all ε>0 and some δ>0,(ii) there exists J≥2 such thatThen for all ε>0Theorem 0.1.2 Let {X_nk,1≤k≤k_n,n≥1} be an array of rowwise negatively associated random variables (if k_n =∞, we will assume that the series Σ_k=1^∞ X_nk converges a.s.) where {k_n,n≥1}(?){1,2,...}∪{∞}, and {c_n, n≥1} a sequence of positive constants such that Σ_n=1^∞c_n=∞. Suppose that for all ε>0 and some δ > 0, Conditions (i), (ii) andhold. ThenMention that assumptions (ii) and (Hi)' of theorem 0.1.2 is somehow difficult to check. For a special case of mean zero array we can establish the following result: Theorem 0.1.3 Let {Xnk,\\} be an array of rowwise negativelyassociated random variables (if kn =00, we will assume that the series Xt-i^* converges a.s.) where {An,w>l}c{l,2,...}U{°°}, and {cn, n>\) a sequence of positive constants such that ^°°_lcn =00. Let q>(x) be a real function such that forsome S > 0: swpx>s x I 0:(II) there exists J>2 such that ￡I,c?(Ew￡M ^l))<00'(III) max^ I XL W\ X* I) l"> 0, as n -* qo .Then E!!lic-^max^lZHi^-*l>^<00 fora11 e>0-Chapter 2 Precise rates in the law of logarithm for positively associated random variablesSince Esary et al. introduced the concept of positively associated random variables, many authors have studied this concept and proved interesting results. Hsu and Robbins introduced the concept of complete convergence in 1947. Precise asymptotics are extensions of complete convergence. In this chapter, we extend the results of Fu KeAng^22' to the positively associated sequences:Theorem 0.2.1 Suppose that an = 0(1 / log n). Then for any b > -1, we haveandwhere //2(A+1) is the 2{b + l)th absolute moment of the standard normal distributions. Theorem 0.2.2 For any 6>-l,wehave— ^— /?>#? ■ i\2*+3Theorem 0.23 Suppose that an = 0(1 / log ri). Then for any b > -1, we have//2(A+1)limff￡/(| ^ ^ (sH)ff^)where ^2(i+1) is the 2(Z> + l)th absolute moment of the standard normal distributions and 2t(￡) = {n :| 5? |> (e + an)\} be a sequence of 0?-mixing positive square integrable random variables, and it satisfies // = EX\ > 0 , E \ X\\\p< oo (p > 2), ￡"=y 2(2') < °o .Denote T^^'^-u), cr2n=ETn2. cr2n=ET^co as ?->?.Theorem 0.3.1 Let (Xki)j=l2..k,k = \,2,--- be a triangular array satisfies the following conditions: every row (Xkx,Xk2,---,Xkk) is an independent copy of (Xi,X2,---,Xk), but every row is independent. Denote Sk=Xkl+Xk2+-uk uuk u1T^logflWhere AT is a standard normal rv.