The Annals of Applied Probability

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

Mireille Bousquet-Mélou and Svante Janson

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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.

Article information

Ann. Appl. Probab., Volume 16, Number 3 (2006), 1597-1632.

First available in Project Euclid: 2 October 2006

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C15
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05C05: Trees

Random binary tree natural labeling vertical profile ISE local limit law


Bousquet-Mélou, Mireille; Janson, Svante. The density of the ISE and local limit laws for embedded trees. Ann. Appl. Probab. 16 (2006), no. 3, 1597--1632. doi:10.1214/105051606000000213.

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