Almost sure central limit theorem for self-normalized products of the some partial sums of ρ−-mixing sequences

Let { X , X n } n ∈ N be a strictly stationary ρ − -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 = 1 k S k , i ( k − 1 ) n μ n ) μ β V k , where β > 0 is a constant, and E ( X ) = μ , S k , i = ∑ j = 1 k X j − X i , 1 ≤ i ≤ k , V k 2 = ∑ i = 1 k ( X i − μ ) 2 .