Abstract
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) Hölder 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-Mélou 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.
Citation
Mireille Bousquet-Mélou. Svante Janson. "The density of the ISE and local limit laws for embedded trees." Ann. Appl. Probab. 16 (3) 1597 - 1632, August 2006. https://doi.org/10.1214/105051606000000213
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