Invairance Principle And Laws Of The Iterated Logarithm For Sums Of A Function Of Trimmed Sums

Probability limit theory is one of the important branches of probability theory. Studyon the limit properties of the product of sums and trimmed sums, are the important themeof the limit theory.Since1998, after Arnold and Villasen or[3]obtained a limit theorem about productof sums of random variables. Many scholars improved on asymptotic properties of theproducts of sums and trimmed sums. Liu Wei-dong and Lin Zheng-yan[6]showed thatthe self-normalized random products of sums for mixing sequences is asymptotically log-normal; Zhang Li-xin and Huang Wei[7], Matula and Stepien′[8]discussed about weakconvergence of partial sums; Wang Fang and Cheng Shi-hong[15]obtained an almost surecentral limit theorem for heavily trimmed sums; under the additional condition that thedistribution is general medium tailed, Zou Hai-lian and Zhang Li-xin[16]proved that theasymptotic distribution of the product of trimmed sums is lognormal; Zang Qing-pei and Lin Zheng-yan discussed the asymptotic distribution of the random product of trimmed sums; with using the weak convergence theorem and continuous mapping theorem, Zhou Rui and Yang Jin-ying proved the invariance principle for the product of trimmed sums and so on. In this paper, based on the existing conclusions, we discuss and promote the asymptotic properties of a function of trimmed sums.The first part (the first chapter), it gives the definitions and the domestic and foreign research status of partial sums and trimmed sumsThe second part (the second chapter), it gives the asymptotic distributions for prod-ucts of sums and trimmed sums of the random variable sequence. Based on the existing conclusions, the invariance principle for the product of trimmed sums was extended to the function of trimmed sums:Theorem2.6Let{Xn,n≥1} be a sequence of i.i.d. random variables with continuous distribution in the domain of attraction of stable law Sa(σ,β,μ) with α∈(1,2],μ=EX1, and the distribution is general medium tailed. Assume that f is a real function defined on an interval I such that P(X1∈I)=1and f’(μ) exists. Then, as n�'∞The third part (the third chapter), it gives the laws of the iterated logarithm for prod-ucts of sums. Based on the existing conclusions, the laws of the iterated logarithm was extended to the function of trimmed sums:Theorem3.5Let{Xk,k≥1} be a sequence of i.i.d. random variables. EX1=μ, VarX1=σ2<∞. Assume that f is a real function defined on an interval I such that P(X1∈I)=1and f’(μ) exists, let Then with probability one is relative compact and the limit set isxf (μ)g(u)0udu, g∈G,0≤x≤1.In particularnf(Tk(a)√lim sup√k) f(μ)n�'∞k=1σ2n log log n=2f (μ) a.s..And then, it gives the following corollary:Corollary3.6Under the condition of theorem3.5, and {Xk} is positive sequences ofrandom variables, let f(x)=log x, then with probability one√1/(γ2n log log n)∞[nt]Tk(a)kμ,≤t≤1k=1n=3is relative compact and the limit set isxexpg(u)du, g∈G,0≤x≤1.0uIn particular√n1/(γ2n log log n)lim supTk(a)√n�'∞k=1kμ=e2a.s..