  R. Doney identifies a striking factorization of the arc-sine law in terms of
the suprema of two independent stable processes of the same index by an elegant
random walks approximation. In this paper, we provide an alternative proof and
a generalization of this factorization based on the theory recently developed
for the exponential functional of L\'evy processes. As a by-product, we provide
some interesting distributional properties for these variables and also some
new examples of the factorization of the arc-sine law.
