Central Limit Theorem For The Sum Of Partial Sum Of Nonstationary NA Sequence

Central Limit Theorem for The Sum of Partial Sum of Nonstationary NA SequenceProbability Limit Theory is one of the important branches and also an essential the oritical foundations of science of Probability and Statistics. Kolmogrov, a famous probability scholar from previous Soviet Union said :"Only Probability Limit Theory can reveal the epistemological value of Probability . Without it,you couldn't understand the real meaning of the fundamental conceptions in Probability." Central Limit Theorem is the signify achievement in the process of Probability. In this paper, we also focus on the CLT and discuss Central Limit Theorem for the sum ofpartialsum of nonstationary NA sequenceIn the 1980's, the concept of negatively associated random variables, which include the concepet of independent random varaibles, was introduced by Alam and Saxena(1981),Blok,Savits and Shaked(1982),Joag-Dev and Proschan (1983) and Joag-Dev (1983).Definition A A finite family of random variables {X_i, 1≤i≤n} are said to be Negatively Associated(NA in short) if for every pair of disjoint subsets A₁, A₂ of {1,2,…, n},Cov(f₁(X_i, i∈A₁), f₂(X_i, i∈A₂))≤0,whenever f₁ and f₂ are real coordinatewise incressing(or decresing)and the covariance exists. An infinite family is Negatively Associated if every finite subfamily is Negatively Associated. Since the concept of NA random varables has a lot of applications in 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.Central Limit Theorem is a significant part of Probability Limit Theory,the investigation for CLT has been always important in the reaserch of Limit Theory to the scholars at home and abroad, so is the NA sequence. Newman(1984)gave a comprehensive description for the reserch results about the Limit Theorem for PA and NA ramdom variable sequences, including the Central Limit Theorem for stationary NA sequences. Su Chun, Chi Xiang(1998) removed the restriction of "stationariness", obtained the Central Limit Theroem for non-stationary identically distributed NA sequences, and Zhang Lixin(2000)remove the restriction of "identical distribution" in the theoremaboved.As we all konw, for a long time, we were established in the investigation of the limit properties of S_n, which is the partial sum of ramdom variables, but for T_n =∑_i=1ⁿ S_i, the sum of partial sum of ramdom variables, there are few results.In fact, it is necessary to investgate the limit properties of T_n, not noly in theory but also in practice. Resnick(1973) and Arnold (1998)found that they should reserch T_n when they were investgating the Limit Throey of record values:{X_n,n≥1} is a random variables sequence, and {Xⁿ,n≥1} denotes the record values sequence, if {X_n, n≥1} is a sequence of i.i.d random variableswith exponential distribution(λ= 1), then Xⁿ (?)∑_i=1ⁿ S_i, so they put forward it is necessary to investgate the limit properties of T_n. In practical problems, such as random walk, ruin theory and time series theory, the investgationof T_n is also necessary. For example, random walk {S_t,t = 0,1,2,…}, where S₀ = 0 and S_t =∑_i=1^t X_i, then S_t can be modeled as the closing price of sime securities(stock) on tth day, and∑_t=1ⁿ S_t/n is the n-th day moving average.According to this, Jiang Tao and Lin Riqi(2002) investgated the Limit Theorem of sum of partial sum of i.i.d. random variables sequences(T_n), they obtained some same covergence condition with the partial sum (S_n), including Central Limit Theorem as follows:Theorem A Let {X_n, n≥1} be a sequence of i.i.d. random variables,Yu Shihang(2004), Yu Shihang, Zhang Ruimei(2007)obtained the results for NA sequences.Theorem B Let {X_n,n≥1} be a sequence of stationary NA random variables, if(A₁) EX₁ = 0, EX₁² <∞;(A₂)σ² = EX₁² + 2∑_i=2^∞EX₁X_i > 0;(A₃) there exists positive integer p and q such that p + q≤n when n islarge enough, and q/p→0,(?)|Cov(X_i,X_r+1|→0;(A₄) (?)δ> 0,such that E|X₁|²⁺⁶ <∞. thenIn this paper, we will remove the the restriction of "stationariness" and obtain some results for non-stationary NA sequences:Theorem 1 Let {X_n, n≥1} be a sequence of NA random variables, if(A₁)EX_i = 0,(?)n≥1:(A₂) (?) (T_n²)/(n³) =σ² > 0; (A4) there exists a random variable X withi EX² <∞some constant C > 0, such that P(|X_i|≥x)≤CP(|X|≥x),(?)x > 0,(?)i≥1. thenIn many pratical problems, random sum is more significant than nonrandomsum, for example, the aggregate claim (?)X_i(where N(t) is the numberof claims up to time t)in ruin theory and geometry sum (?)X_i (where v isa rundom variable with geometry distribution)are both random sum.Renyi(1960), Blum (1963) and Mogyorodi(1962) investigated the Central Limit Theorem for random sum of i.i.d. random variables sequence, Jiang Tao, Su Chun, Tang Qihe(2001) investigated the Central Limit Theorem for random sum of partial sum of i.i.d. random variables sequence, they obtained the results as follows:Theorem C Let {X_n,n≥1} be a sequence of i.i.d. random variables with EX₁ = 0 and EX₁² = 1, {Ï„_n, n≥1} is such a sequence of positive integervaluedrandom variables thatÏ„_n/b_n (?) c, where constant c > 0, and b_n is a positive integer-valued number with b_n↑∞, thenInspired by this, in the paper, we also extende our result to the randomsum of partial sum of nonstationary NA sequence, and the corresponding theorem as follows:Theorem 2 Let {X_n,n≥1} be a sequence of NA random variables satisfying the conditions (A₁), (A₂), (A₃) and (A₄) in Theorem 1, {Ï„_n,n≥1} is such a sequence of positive integer-valued random variables thatÏ„_n/b_n (?) Ï„, whereÏ„is a positive random variable, and b_n is a positive integer-valued number with b_n↑∞, if there exisits N₀∈N such thatφ_N0 = 1, thenLet {X_n,n≥1} be a sequence of random variables,σ-algebra F_a,b =σ{X_i, a≤i≤b}, -∞< a≤b < +∞, then for k∈N, we give the definition ofφ_k sa follows:Definition Aφ_k = inf{(P(A∩B))/(P(A)P(B): A∈F_1,l, B∈F_1+k,+∞, P(A)P(B)...