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2020 Characterising random partitions by random colouring
Jakob E. Björnberg, Cécile Mailler, Peter Mörters, Daniel Ueltschi
Electron. Commun. Probab. 25: 1-12 (2020). DOI: 10.1214/19-ECP283

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 $\nicefrac {1}{2}$.

Citation

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Jakob E. Björnberg. Cécile Mailler. Peter Mörters. Daniel Ueltschi. "Characterising random partitions by random colouring." Electron. Commun. Probab. 25 1 - 12, 2020. https://doi.org/10.1214/19-ECP283

Information

Received: 13 July 2019; Accepted: 21 December 2019; Published: 2020
First available in Project Euclid: 13 January 2020

zbMATH: 1439.60047
MathSciNet: MR4053907
Digital Object Identifier: 10.1214/19-ECP283

Subjects:
Primary: 60E10, 60G57, 60K35

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