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

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



Let (X n ) be a sequence of i.i.d random variables and U n a U-statistic corresponding to a symmetric kernel function h, where h 1(x 1) = Eh(x 1, X 2, X 3, . . . , X m ), μ = E(h(X 1, X 2, . . . , X m )) and ς 1 = Var(h 1(X 1)). Denote \({\gamma=\sqrt{\varsigma_{1}}/\mu}\), the coefficient of variation. Assume that P(h(X 1, X 2, . . . , X m ) > 0) = 1, ς 1 > 0 and E|h(X 1, X 2, . . . , X m )|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.


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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