A connection between self-normalized products and stable laws

Let X1,…,XnX1,…,Xn constitute a random sample from a population with underpinning cumulative distribution function F(x)F(x). For any value 0<α<10<α<1, we prove that under a condition of stable laws, the self-normalized product n1/2αX1X2…Xn/∑*Xi12…Xin-12 follows the same distribution as X1X1, where ∑*∑* denotes the sum of over all permissible sequences of integers 1⩽i1