An almost sure central limit theorem for self-normalized products of sums of i.i.d. random variables

Let X , X 1 , X 2 , … be a sequence of independent and identically distributed positive random variables with EX = μ > 0 . In this paper we show that the almost sure central limit theorem for self-normalized products of sums holds only under the assumptions that X belongs to the domain of attraction of the normal law.