The density of the ISE and local limit laws for embedded trees

• 成果类型: Article
• 署名作者: Bousquet-Melou, Mireille; Janson, Svante
• 署名单位: Universite de Bordeaux; Centre National de la Recherche Scientifique (CNRS); Uppsala University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051606000000213
• 发表日期: 2006
• 页码: 1597-1632
• 关键词: partial-differential equations; continuum random tree; diffusion; AREA

摘要

It has been known for a few years that the occupation measure of several models of embedded trees converges, after a suitable normalization, to the random measure called ISE (integrated SuperBrownian excursion). Here, we prove a local version of this result: ISE has a (random) Milder continuous density, and the vertical profile of embedded trees converges to this density, at least for some such trees. As a consequence, we derive a formula for the distribution of the density of ISE at a given point. This follows from earlier results by Bousquet-Melou on convergence of the vertical profile at a fixed point. We also provide a recurrence relation defining the moments of the (random) moments of ISE.