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

Abstract

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.