# Independence by Random Scaling

## TL;DR

This paper establishes conditions for coupling a scalar random variable with a scaled version to achieve independence, extending known results and applying to Poisson-Dirichlet distributions and diffusion excursions.

## Contribution

It provides new conditions for independence via random scaling and generalizes existing results to negative parameter ranges and diffusion processes.

## Key findings

- Conditions for independence coupling of T and ξT
- Generalization of Pitman-Yor result to negative parameters
- Application to diffusion excursions and exponential times

## Abstract

We give conditions under which a scalar random variable T can be coupled to a random scaling factor $\xi$ such that T and $\xi$T are rendered stochastically independent. A similar result is obtained for random measures. One consequence is a generalization of a result by Pitman and Yor on the Poisson-Dirichlet distribution to its negative parameter range. Another application are diffusion excursions straddling an exponential random time.