Spatial central limit theorem for supercritical superprocesses
type: Item
instance of: scholarly article
title: Spatial central limit theorem for supercritical superprocesses
zbMATH Open document ID: 1391.60045
author: Piotr Miłoś
published in: Journal of Theoretical Probability
publication date: 2018-04-20T00:00:00Z
review text: A measure-valued diffusion (a superprocess) \(X_t\) can be seen as the short lifetime and high-density limit of a branching particle system. It is described by a couple \((L,\psi)\), where \(L\) is the infinitesimal generator of a strongly recurrent diffusion on \(\mathbb{R}^d\) and \(\Psi\) is a branching mechanism. If \(\varphi\) denotes the unique invariant measure of the diffusion generated by \(L\), a spatial law of large numbers for \(X_t\) has been proved in [\textit{J. Engländer} and \textit{A. Winter}, Ann. Inst. Henri Poincaré, Probab. Stat. 42, No. 2, 171--185 (2006; Zbl 1093.60058)], namely \[ \lim_{t \to \infty} e^{-\alpha t} \langle X_t, f\rangle = \langle \varphi, f\rangle V_\infty \] for any bounded continuous function \(f\). Here \(\alpha\) is the asymptotic growth rate of the system and \(V_\infty:=\lim_{t \to \infty} e^{-\alpha t} |X_t|\) is a non-trivial random variable. Here \(|X_t|=\langle X_t, 1\rangle\). The paper at hand provides a corresponding central limit theorem, i.e., convergence in distribution of \[ \frac{\langle X_t , f \rangle - |X_t| \langle \varphi, f \rangle}{N_t}. \] Three qualitatively different regimes (with different normalizations \(N_t\)) occur, depending on whether \(\alpha < 2 \mu\); \(\alpha=2 \mu\) or \(\alpha > 2 \mu\), where \(\mu\) is the exponential rate of convergence of a diffusion generated by \(L\) towards stationarity. This corresponds to the interplay of the growth governed by \(\psi\) and the spatial evolution governed by \(L\). In the first two regimes, Gaussian limits occur; while the limit in the third regime is conjectured to be non-Gaussian. Also, joint convergence of the spatial fluctuations and fluctuations of the total mass is provided, showing that they become asymptotically independent.
reviewed by: Sebastian Mentemeier
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G20
zbMATH DE Number: 6862616
zbMATH Keywords: branching processes
zbMATH Keywords: supercritical branching processes
zbMATH Keywords: limit behavior
zbMATH Keywords: central limit theorem
MaRDI profile type: Publication
full work available at URL: s10959-016-0704-6
OpenAlex ID: W2476995849
Wikidata QID: Q59462032
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DOI: 10.1007/S10959-016-0704-6
description: scientific article; zbMATH DE number 6862616