A note on the almost sure central limit theorem for the product of some partial sums (Q2258641)

Based on the authors' abstract: A sequence \((X_n)\) of i.i.d., positive, square integrable random variables with \(\operatorname{E}(X_1)= \mu>0\), \(\text{Var}(X_1)=\sigma^2\) is considered. Denote by \(S_{n,k}=\sum^n_{i=1} X_i- X_k\) and by \(\gamma= \sigma/\mu\) the coefficient of variation. The aim of this paper is to show the existence of an unbounded measurable function \(g\) that satisfies the almost sure central limit theorem: \[ \lim_{N\to\infty} {1\over\log N} \sum^N_{n=1} {1\over n} g\Biggl(\Biggl({\prod^n_{k= 1} S_{n,k}\over (n-1)^n \mu^n}\Biggr)^{{1\over\gamma\sqrt{n}}}\Biggr)= \int^\infty_0 g(x)dF(x)\quad\text{a.s.}, \] where \(F(.)\) is the distribution function of the random variable \(e^{{\mathcal N}}\) and \({\mathcal N}\) is a standard normal random variable.