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 ? 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, ? ₁ > 0 and E|h(X ₁, X ₂, . . . , X _m )|³ < ∞. We give herein the conditions under which

$lim_{Nrightarrowinfty}frac{1}{log N}sum_{n=1}^{N}frac{1}{n}gleft(left(prod_{k=m}^{n}frac{U_{k}}{mu}right)^{frac{1}{mgammasqrt{n}}}right) =intlimits_{-infty}^{infty}g(x)dF(x)quad {rm a.s.}$

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.