Metrika

, Volume 73, Issue 1, pp 61–76 | Cite as

Almost sure central limit theorem for the products of U-statistics

Article

Abstract

Let (Xn) be a sequence of i.i.d random variables and Un a U-statistic corresponding to a symmetric kernel function h, where h1(x1) = Eh(x1, X2, X3, . . . , Xm), μ = E(h(X1, X2, . . . , Xm)) and ς1 = Var(h1(X1)). Denote \({\gamma=\sqrt{\varsigma_{1}}/\mu}\), the coefficient of variation. Assume that P(h(X1, X2, . . . , Xm) > 0) = 1, ς1 > 0 and E|h(X1, X2, . . . , Xm)|3 < ∞. We give herein the conditions under which
$$\lim_{N\rightarrow\infty}\frac{1}{\log N}\sum_{n=1}^{N}\frac{1}{n}g\left(\left(\prod_{k=m}^{n}\frac{U_{k}}{\mu}\right)^{\frac{1}{m\gamma\sqrt{n}}}\right) =\int\limits_{-\infty}^{\infty}g(x)dF(x)\quad {\rm a.s.}$$
for a certain family of unbounded measurable functions g, where F(·) is the distribution function of the random variable \({\exp(\sqrt{2} \xi)}\) and ξ is a standard normal random variable.

Keywords

Almost sure central limit theorem U-statistics Unbounded measurable function 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Zuoxiang Peng
    • 1
  • Zhongquan Tan
    • 2
  • Saralees Nadarajah
    • 3
  1. 1.School of Mathematics and StatisticsSouthwest UniversityChongqingPeople’s Republic of China
  2. 2.Department of MathematicsZunyi Normal CollegeZunyiPeople’s Republic of China
  3. 3.School of MathematicsUniversity of ManchesterManchesterUK

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