Acta mathematica scientia,Series A ›› 2013, Vol. 33 ›› Issue (3): 475-482.

• Articles • Previous Articles     Next Articles

The Asymptotic Properties and Almost Sure Central Limit Theorems for the Products of a Class of Statistics

 QIU Jin1, LU Chuan-Rong2   

  1. 1.School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou 310018;
    2.Department of Mathematics, Zhejiang University, Hangzhou 310028
  • Received:2011-11-19 Revised:2013-02-28 Online:2013-06-25 Published:2013-06-25
  • Supported by:

    浙江省自然科学基金(Y6110615)和教育部人文社会科学研究规划基金(12YJA910003)资助

Abstract:

Let {Xn, -∞<n<∞} be a sequence of independent and identically distributed, positive, square integrable random variables with μ=EX1σ2=Var X1>0. The asymptotic properties for the products of a class of statistics (or random functions) expressed by Tn=anSn+Rn are discussed, where Sn=∑ni=1Xi, an>0 is a sequence of constants, Rn=o(ann) a.s.. The results contain the almost sure central limit theorems, asymptotically lognormality and the weak invariance principles. Some examples such as U-statistics, Von-Mises statistics, error variance estimates in linear models are stated to illustrate the generality of the results.

Key words: Products of statistics, Almost sure central limit theorem, Central limit theorem, Weak invariance principle

CLC Number: 

  • 60F05
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