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

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.