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

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

2005-09-14

arxiv.org/abs/math/0509322v2

The Annals of Applied Probability 16, 3 (2006) 1597--1632

Mathematics

Probability

Scientific paper

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) H\"{o}lder 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-M\'{e}lou 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.

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