A note on the almost sure central limit theorem for the product of some partial sums

Let (X-n) be a sequence of i.i.d., positive, square integrable random variables with E(X-1) = mu > 0, Var(X-1) = sigma(2). Denote by S-n,S-k = Sigma X-n(i=1)i - X-k and by gamma = sigma/mu the coefficient of variation. Our goal is to show the unbounded, measurable functions g, which satisfy the almost sure central limit theorem, i.e., lim(N ->infinity)1/logN Sigma=(N)(n=1)1/ng((pi S-n(k=1)n,k/(n - 1)(n)mu(n))(1/gamma root n)) = integral(infinity)(0) g(x)dF(x) a.s., where F(.) is the distribution function of the random variable eN and N is a standard normal random variable.