# Characterising random partitions by random colouring

## TL;DR

This paper investigates how the distribution of a Bernoulli convolution of a random partition of the unit interval can uniquely characterize the partition itself, especially highlighting the special case when p equals 1/2.

## Contribution

It proves that for residual allocation partitions, the Bernoulli convolution uniquely determines the partition distribution unless p equals 1/2.

## Key findings

- Partition distributions are fully characterized by their Bernoulli convolution for p ≠ 1/2.
- The case p=1/2 is exceptional and does not allow unique characterization.
- The results apply specifically to residual allocation partitions.

## Abstract

Let $(X_1,X_2,...)$ be a random partition of the unit interval $[0,1]$, i.e. $X_i\geq0$ and $\sum_{i\geq1} X_i=1$, and let $(\varepsilon_1,\varepsilon_2,...)$ be i.i.d. Bernoulli random variables of parameter $p \in (0,1)$. The Bernoulli convolution of the partition is the random variable $Z =\sum_{i\geq1} \varepsilon_i X_i$. The question addressed in this article is: Knowing the distribution of $Z$ for some fixed $p\in(0,1)$, what can we infer about the random partition? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter $p$ is not equal to $1/2$.