A functional clt for the occupation time of a state-dependent branching random walk

• 成果类型: Article
• 署名作者: Birkner, Matthias; Zaehle, Iljana
• 署名单位: Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; University of Erlangen Nuremberg
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117907000000150
• 发表日期: 2007
• 页码: 2063-2090
• 关键词: limit-theorems; systems; fluctuations; dimensions; range

摘要

We show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension d >= 3, started off either from a Poisson field or in equilibrium, when suitably normalized, converges to a Brownian motion in d >= 4. In d = 3, the limit process is a fractional Brownian motion with Hurst parameter 3/4 when starting in equilibrium, and a related Gaussian process when starting from a Poisson field. For (dependent) branching random walks with state dependent branching rate we obtain convergence in f.d.d. to the same limit process, and for d = 3 also a functional limit theorem.