Almost sure central limit theorem for self-normalized products of the some partial sums of ρ− $\rho^{-}$-mixing sequences

Abstract Let {X,Xn}n∈N $\{X, X_{n}\}_{n\in N}$ be a strictly stationary ρ− $\rho^{-}$-mixing sequence of positive random variables, under the suitable conditions, we get the almost sure central limit theorem for the products of the some partial sums (∏i=1kSk,i(k−1)nμn)μβVk $({\frac{\prod_{i=1}^{k}S_{k,i}}{(k-1)^{n}\mu ^{n}} )^{\frac{\mu}{\beta V_{k}}} }$, where β>0 $\beta>0$ is a constant, and E(X)=μ ${\mathrm{E}}(X)=\mu$, Sk,i=∑j=1kXj−Xi $S_{k,i}=\sum_{j=1}^{k}X_{j}-X_{i}$, 1≤i≤k $1\le i\le k$, Vk2=∑i=1k(Xi−μ)2 $V_{k}^{2}=\sum_{i=1}^{k}(X_{i}-\mu)^{2}$