## The Annals of Applied Probability

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

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 2 October 2006

https://projecteuclid.org/euclid.aoap/1159804993

Digital Object Identifier
doi:10.1214/105051606000000213

Mathematical Reviews number (MathSciNet)
MR2260075

Zentralblatt MATH identifier
1132.60009

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

#### Citation

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. https://projecteuclid.org/euclid.aoap/1159804993

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