The Annals of Applied Probability

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

Mireille Bousquet-Mélou and Svante Janson

Full-text: Open access

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

Permanent link to this document
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

Keywords
Random binary tree natural labeling vertical profile ISE local limit law

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


Export citation

References

  • Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1--28.
  • Aldous, D. (1991). The continuum random tree. II. An overview. Stochastic Analysis. London Math. Soc. Lecture Note Ser. 167 23--70. Cambridge Univ. Press.
  • Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248--289.
  • Aldous, D. (1993). Tree-based models for random distribution of mass. J. Statist. Phys. 73 625--641.
  • Bergh, J. and Löfström, J. (1976). Interpolation Spaces. Springer, Berlin.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Bousquet-Mélou, M. (2006). Limit laws for embedded trees. Applications to the integrated superBrownian excursion. Random Structures Algorithms. To appear.
  • Cohn, D. L. (1980). Measure Theory. Birkhäuser, Boston.
  • Devroye, L. (1998). Branching processes and their applications in the analysis of tree structures and tree algorithms. In Probabilistic Methods for Algorithmic Discrete Mathematics (M. Habib et al., eds.) 249--314. Springer, Berlin.
  • Drmota, M. and Gittenberger, B. (1997). On the profile of random trees. Random Structures Algorithms 10 421--451.
  • Flajolet, P. and Louchard, G. (2001). Analytic variations on the Airy distribution. Algorithmica 31 361--377.
  • Flajolet, P. and Odlyzko, A. (1990). Singularity analysis of generating functions. SIAM J. Discrete Math. 3 216--240.
  • Geman, D. and Horowitz, J. (1980). Occupation densities. Ann. Probab. 8 1--67.
  • Gut, A. (2005). Probability: A Graduate Course. Springer, New York.
  • Janson, S. (2003). The Wiener index of simply generated random trees. Random Structures Algorithms 22 337--358.
  • Janson, S. (2006). Left and right pathlengths in random binary trees. Algorithmica. To appear. Available at http://www.math.uu.se/~svante/papers/.
  • Janson S. and Marckert, J.-F. (2006). Convergence of discrete snakes. J. Theoret. Probab. 18 613--645.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • Konno, N. and Shiga, T. (1988). Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 201--225.
  • Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
  • Le Gall, J.-F. and Weill, M. (2006). Conditioned Brownian trees. Ann. Inst. H. Poincaré Probab. Statist. 42 455--489.
  • Louchard, G. (1984). The Brownian excursion area: A numerical analysis. Comput. Math. Appl. 10 413--417.
  • Marckert, J.-F. (2004). The rotation correspondence is asymptotically a dilatation. Random Structures Algorithms 24 118--132.
  • Marckert, J.-F. and Mokkadem, A. (2003). States spaces of the snake and its tour---convergence of the discrete snake. J. Theoret. Probab. 16 1015--1046.
  • Nguyen The, M. (2004). Area and inertial moment of Dyck paths. Combin. Probab. Comput. 13 697--716.
  • Reimers, M. (1989). One-dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Theory Related Fields 81 319--340.
  • Richard, C. (2005). On $q$-functional equations and excursion moments. Available at http://arxiv.org/abs/math.CO/0503198.
  • Rudin, W. (1991). Functional Analysis, 2nd ed. McGraw--Hill, New York.
  • Stein, E. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press.
  • Sugitani, S. (1989). Some properties for the measure-valued branching diffusion processes. J. Math. Soc. Japan 41 437--462.