## Electronic Journal of Probability

### Brownian Bridge Asymptotics for Random $p$-Mappings

#### Abstract

The Joyal bijection between doubly-rooted trees and mappings can be lifted to a transformation on function space which takes tree-walks to mapping-walks. Applying known results on weak convergence of random tree walks to Brownian excursion, we give a conceptually simpler rederivation of the Aldous-Pitman (1994) result on convergence of uniform random mapping walks to reflecting Brownian bridge, and extend this result to random $p$-mappings.

#### Article information

Source
Electron. J. Probab., Volume 9 (2004), paper no. 3, 37-56.

Dates
Accepted: 10 December 2003
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ejp/1465229689

Digital Object Identifier
doi:10.1214/EJP.v9-186

Mathematical Reviews number (MathSciNet)
MR2041828

Zentralblatt MATH identifier
1064.60012

Rights

#### Citation

Aldous, David; Miermont, Gregory; Pitman, Jim. Brownian Bridge Asymptotics for Random $p$-Mappings. Electron. J. Probab. 9 (2004), paper no. 3, 37--56. doi:10.1214/EJP.v9-186. https://projecteuclid.org/euclid.ejp/1465229689

#### References

• Aldous, David; Pitman, Jim. Two recursive decompositions of Brownian bridge related to the asymptotics of random mappings. Technical Report 595, Dept. Statistics, U.C. Berkeley, 2002.
• Aldous, David. The continuum random tree. III. Ann. Probab. 21 (1993), no. 1, 248–289.
• Aldous, David; Miermont, Gregory; Pitman, Jim. Weak convergence of random $p$-mappings and the exploration process of the inhomogeneous continuum random tree. In preparation, 2004.
• Aldous, David; Pitman, Jim. Brownian bridge asymptotics for random mappings. Random Structures Algorithms 5 (1994), no. 4, 487–512.
• Aldous, David; Pitman, Jim. The asymptotic distribution of the diameter of a random mapping. C. R. Math. Acad. Sci. Paris 334 (2002), no. 11, 1021–1024.
• Aldous, David; Pitman, Jim. Invariance principles for non-uniform random mappings and trees. Asymptotic combinatorics with application to mathematical physics (St. Petersburg, 2001), 113–147, NATO Sci. Ser. II Math. Phys. Chem., 77, Kluwer Acad. Publ., Dordrecht, 2002.
• Bertoin, Jean; Pitman, Jim. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 (1994), no. 2, 147–166.
• Biane, Ph. Relations entre pont et excursion du mouvement brownien réel. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 1, 1–7.
• Biane, Philippe. Some comments on the paper: "Brownian bridge asymptotics for random mappings" [Random Structures Algorithms 5 (1994), no. 4, 487–512] by D. J. Aldous and J. W. Pitman. Random Structures Algorithms 5 (1994), no. 4, 513–516.
• Camarri, Michael; Pitman, Jim. Limit distributions and random trees derived from the birthday problem with unequal probabilities. Electron. J. Probab. 5 (2000), no. 1, 18 pp.
• Drmota, Michael; Gittenberger, Bernhard. On the profile of random trees. Random Structures Algorithms 10 (1997), no. 4, 421–451.
• Harris, B. A survey of the early history of the theory of random mappings. Probabilistic methods in discrete mathematics (Petrozavodsk, 1992), 1–22, Progr. Pure Appl. Discrete Math., 1, VSP, Utrecht, 1993.
• Joyal, André. Une théorie combinatoire des séries formelles. (French) [A combinatorial theory of formal series] Adv. in Math. 42 (1981), no. 1, 1–82.
• Kolchin, Valentin F. Random mappings. Translated from the Russian. With a foreword by S. R. S. Varadhan. Translation Series in Mathematics and Engineering. Optimization Software, Inc., Publications Division, New York, 1986. xiv + 207 pp. ISBN: 0-911575-16-2.
• Marckert, Jean-François; Mokkadem, Abdelkader. The depth first processes of Galton-Watson trees converge to the same Brownian excursion. Ann. Probab. 31 (2003), no. 3, 1655–1678.
• O'Cinneide, C. A.; Pokrovskii, A. V. Nonuniform random transformations. Ann. Appl. Probab. 10 (2000), no. 4, 1151–1181.
• Pitman, Jim. Random mappings, forests, and subsets associated with Abel-Cayley-Hurwitz multinomial expansions. Sém. Lothar. Combin. 46 (2001/02), Art. B46h, 45 pp.
• Pitman, Jim. Combinatorial Stochastic Processes. Technical Report 621, Dept. Statistics, U.C. Berkeley, 2002. Lecture notes for St. Flour course, July 2002.
• Pitman, Jim; Yor, Marc. Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. (3) 65 (1992), no. 2, 326–356.