Distribution of Symmetric Stable Laws of Index $2^{-n}$

Abstract

Let $X_1, X_2 \cdots, X_n(n \geq 2)$ be independent standard normal. Then the random variable $U = X_1/V_n$ where $V_n = \exp_2\lbrack 2^{n - 2} - 1\rbrack X_2(X_3)^2 \cdots(X^2_n)^{2^{n - 3}} \quad n \geq 3 \\ = X_2 \text{for} n = 2$ has a symmetric stable distribution with index $2^{2 - n}$.