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Infinitely ramified point measures and branching Lévy processes. (English) Zbl 1456.60225
In analogy to the well-known relation between infinitely divisible distributions and processes with stationary independent increments (Lévy processes), the authors connect what they call infinitely ramified point measures (IRPM) with branching Lévy processes (BLP). An IRPM is defined as a random point measure \(\mathcal{Z}\) which for every \(n\in\mathbb{N}\) has the same distribution as the \(n\)th generation of some branching random walk. In the considered BLP, particles move independently according to Lévy process and produce progeny during their lifetime similarly as in a Crump-Mode-Jagers branching process. The point measures, random walks and Lévy processes are taken here on the real line. Denote \(\langle\Sigma_n \delta_{x_n},f\rangle:= \Sigma_n f(x_n)\) and \(\textbf{e}_\theta(x) :=\textbf{e}^{x\theta}\), \(\theta\ge 0\), \(x\in\mathbb{R}\). It is shown that given an IRPM \(\mathcal{Z}\) such that (*) \(0< E(\langle\mathcal{Z}, \textbf{e}_\theta\rangle)<\infty\) for some \(\theta\ge 0\), there exists a BLP \(Z= \{Z_t; t\ge 0\}\) with \(\mathcal{Z}\overset{(d)}{=} Z_1\). Vice versa, if \(Z\) is a BLP such that the corresponding Lévy measure satisfies certain integrability conditions, then \(Z_1\) is an IRPM satisfying (*).

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G51 Processes with independent increments; Lévy processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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