Characterising random partitions by random colouring

Abstract

Let $(X_{1},X_{2},...)$ be a random partition of the unit interval $[0,1]$, i.e. $X_{i}\geq 0$ and $\sum _{i\geq 1} X_{i}=1$, 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 $Z =\sum _{i\geq 1} \varepsilon _{i} X_{i}$. 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 $(X_{1}, X_{2},...)$? 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 $\nicefrac {1}{2}$.