# On Doney's striking factorization of the arc-sine law

## TL;DR

This paper offers an alternative proof and generalization of Doney's factorization of the arc-sine law, using Lévy process theory, and explores new distributional properties and examples related to this factorization.

## Contribution

It introduces a new proof and broader generalization of Doney's arc-sine law factorization using exponential functionals of Lévy processes.

## Key findings

- New distributional properties of the variables involved
- Additional examples of the arc-sine law factorization
- Alternative proof of Doney's original result

## Abstract

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.