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Characterising random partitions by random colouring. (English) Zbl 1439.60047

Summary: Let \((X_1,X_2,\ldots)\) 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 \((\varepsilon_1, \varepsilon_2\ldots)\) 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 \((X_1, X_2,\ldots)\)? 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 \(\frac{1} {2}\).

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60E05 Probability distributions: general theory
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