Characterising random partitions by random colouring

Jakob Björnberg, Cecile Mailler, Peter Mörters, Daniel Ueltschi

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Abstract

Let (X1, X2, …) be a random partition of the unit interval [0, 1], i.e. Xi ≥ 0 and ∑i≥1 Xi = 1, and let (ε1, ε2, …) be i.i.d. Bernoulli random variables of parameter p ϵ (0, 1). The Bernoulli convolution of the partition is the random variable Z =∑i≥1εiXi . The question addressed in this article is: Knowing the distribution of Z for some fixed p ϵ (0, 1), what can we infer about the random partition (X1, X2, …)? 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 to1/2.
Original languageEnglish
Article number4
JournalElectronic Communications in Probability
Volume25
DOIs
Publication statusPublished - 2020

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