ASYMPTOTIC FLUCTUATIONS IN SUPERCRITICAL CRUMP-MODE-JAGERS PROCESSES

• 成果类型: Article
• 署名作者: Iksanov, Alexander; Kolesko, Konrad; Meiners, Matthias
• 署名单位: Ministry of Education & Science of Ukraine; Taras Shevchenko National University of Kyiv; University of Wroclaw; Justus Liebig University Giessen
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1697
• 发表日期: 2024
• 页码: 1538-1606
• 关键词: markov branching-processes; LIMIT-THEOREMS; CONVERGENCE

摘要

Consider a supercritical Crump-Mode-Jagers process (Z(t)(phi))(t >= 0) counted with a random characteristic phi. Nerman's celebrated law of large numbers [Z. Wahrsch. Verw. Gebiete 57, 365--395, 1981] states that, under some mild assumptions, e(-alpha t)Z(t )(phi)converges almost surely as t ->infinity to aW. Here, alpha>0 is the Malthusian parameter, a is a constant and W is the limit of Nerman's martingale, which is positive on the survival event. In this general situation, under additional (second moment) assumptions, we prove a central limit theorem for (Z(t)(phi))(t >= 0). More precisely, we show that there exist a constant k is an element of N(0 )and a function H(t), a finite random linear combination of functions of the form t(j)e(lambda t) with alpha/2 <= Re(lambda)