Electronic Communications in Probability

Characterising random partitions by random colouring

Jakob E. Björnberg, Cécile Mailler, Peter Mörters, and Daniel Ueltschi

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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 $\nicefrac {1}{2}$.

Article information

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

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

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Zentralblatt MATH identifier

Primary: 60E10: Characteristic functions; other transforms 60G57: Random measures 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

partition structures residual allocation

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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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