## Electronic Communications in Probability

### 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 $\frac{1} {2}$.

#### Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 4, 12 pp.

Dates
Accepted: 21 December 2019
First available in Project Euclid: 13 January 2020

https://projecteuclid.org/euclid.ecp/1578906086

Digital Object Identifier
doi:10.1214/19-ECP283

#### Citation

Björnberg, Jakob E.; Mailler, Cécile; Mörters, Peter; Ueltschi, Daniel. Characterising random partitions by random colouring. Electron. Commun. Probab. 25 (2020), paper no. 4, 12 pp. doi:10.1214/19-ECP283. https://projecteuclid.org/euclid.ecp/1578906086

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