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Occupation densities of ensembles of branching random walks. (English) Zbl 1434.60243
Summary: We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of \(\lfloor sN\rfloor\) branching random walks, viewed as a function-valued, increasing process \(\{g_s^N\}_{s\ge 0}\), converges weakly to a pure jump process in the Skorohod space \(\mathbb{D} ([0, +\infty), \mathcal{C}_0(\mathbb{R}))\), as \(N\to \infty \). Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-valued stable-1/2 subordinator.
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G50 Sums of independent random variables; random walks
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