## Electronic Journal of Probability

### Asymptotic properties of expansive Galton-Watson trees

#### Abstract

We consider a super-critical Galton-Watson tree $\tau$ whose non-degenerate offspring distribution has finite mean. We consider the random trees $\tau _n$ distributed as $\tau$ conditioned on the $n$-th generation, $Z_n$, to be of size $a_n\in{\mathbb N}$. We identify the possible local limits of $\tau _n$ as $n$ goes to infinity according to the growth rate of $a_n$. In the low regime, the local limit $\tau ^0$ is the Kesten tree, in the moderate regime the family of local limits, $\tau ^\theta$ for $\theta \in (0, +\infty )$, is distributed as $\tau$ conditionally on $\{W=\theta \}$, where $W$ is the (non-trivial) limit of the renormalization of $Z_n$. In the high regime, we prove the local convergence towards $\tau ^\infty$ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits $(\tau ^\theta , \theta \in [0, \infty ])$.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 15, 51 pp.

Dates
Received: 12 December 2017
Accepted: 30 January 2019
First available in Project Euclid: 22 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1550826098

Digital Object Identifier
doi:10.1214/19-EJP272

Mathematical Reviews number (MathSciNet)
MR3916335

Zentralblatt MATH identifier
07055653

#### Citation

Abraham, Romain; Delmas, Jean-François. Asymptotic properties of expansive Galton-Watson trees. Electron. J. Probab. 24 (2019), paper no. 15, 51 pp. doi:10.1214/19-EJP272. https://projecteuclid.org/euclid.ejp/1550826098

#### References

• [1] R. ABRAHAM, A. BOUAZIZ, and J.-F. DELMAS, Local limit of fat geometric Galton-Watson trees, arXiv:1709.09403, 2017.
• [2] R. ABRAHAM and J.-F. DELMAS, An introduction to Galton-Watson trees and their local limits, arXiv:1506.05571, 2015.
• [3] R. ABRAHAM and J.-F. DELMAS, Local limits of conditioned Galton-Watson trees: the condensation case, Elec. J. of Probab. 19 (2014), Article 56, 1–29.
• [4] R. ABRAHAM and J.-F. DELMAS, Local limits of conditioned Galton-Watson trees: the infinite spine case, Elec. J. of Probab. 19 (2014), Article 2, 1–19.
• [5] G. ALSMEYER and U. RÖSLER, The Martin entrance boundary of the Galton-Watson process, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 5, 591–606.
• [6] S. ASMUSSEN and H. HERING, Branching processes, Progress in Probability and Statistics, vol. 3, Birkhäuser Boston, Inc., Boston, MA, 1983.
• [7] K. ATHREYA and P.E. NEY, The local limit theorem and some related aspects of super-critical branching processes, Trans. Amer. Math. Soc. 152 (1970), 233–251.
• [8] K. ATHREYA and P.E. NEY, Branching processes, Springer-Verlag, New York-Heidelberg, 1972, Die Grundlehren der mathematischen Wissenschaften, Band 196.
• [9] N. BERESTYCKI, N. GANTERT, P. MÖRTERS, and N. SIDOROVA, Galton-Watson trees with vanishing martingale limit, J. Stat. Phys. 155 (2014), no. 4, 737–762.
• [10] J. D. BIGGINS and N. H. BINGHAM, Large deviations in the supercritical branching process, Adv. in Appl. Probab. 25 (1993), no. 4, 757–772.
• [11] H. COHN, Harmonic functions for a class of Markov chains, J. Austral. Math. Soc. Ser. A 28 (1979), no. 4, 413–422.
• [12] S. DUBUC and E. SENETA, The local limit theorem for the Galton-Watson process, Ann. of Probab. 4 (1976), 490–496.
• [13] S. DUBUC, Problèmes relatifs à l’itération de fonctions suggérés par les processus en cascade, Ann. Inst. Fourier (Grenoble) 21 (1971), no. 1, 171–251.
• [14] S. DUBUC, États accessibles dans un processus de Galton-Watson, Canad. Math. Bull. 17 (1974), 111–113.
• [15] E. B. DYNKIN, Boundary theory of Markov processes (discrete case), Russian Mathematical Surveys 24 (1969), no. 2 (146), 1–42.
• [16] P. FLAJOLET and A. M. ODLYZKO, Limit distributions for coefficients of iterates of polynomials with applications to combinatorial enumerations, Math. Proc. Cambridge Philos. Soc. 96 (1984), no. 2, 237–253.
• [17] K. FLEISCHMANN and V. WACHTEL, Lower deviation probabilities for supercritical Galton-Watson processes, Ann. Inst. H. Poincaré 43 (2007), no. 2, 233–255.
• [18] K. FLEISCHMANN and V. WACHTEL, On the left tail asymptotics for the limit law of supercritical Galton-Watson processes in the Böttcher case, Ann. Inst. H. Poincaré 45 (2009), no. 1, 201–225.
• [19] B. M. HAMBLY, On constant tail behaviour for the limiting random variable in a supercritical branching process, J. Appl. Probab. 32 (1995), no. 1, 267–273.
• [20] T. E. HARRIS, Branching processes, Ann. Math. Statistics 19 (1948), 474–494.
• [21] S. JANSON, Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation, Probab. Surv. 9 (2012), 103–252.
• [22] T. JONSSON and S.O. STEFANSSON, Condensation in nongeneric trees, J. Stat. Phys. 142 (2011), 277–313.
• [23] J. G. KEMENY, J. L. SNELL, and A. W. KNAPP, Denumerable Markov chains, second ed., Springer-Verlag, New York-Heidelberg-Berlin, 1976, With a chapter on Markov random fields, by David Griffeath, Graduate Texts in Mathematics, No. 40.
• [24] H. KESTEN, Subdiffusive behavior of random walk on a random cluster, Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 4, 425–487.
• [25] Q. LIU, Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks, Stochastic Process. Appl. 82 (1999), no. 1, 61–87.
• [26] J. C. LOOTGIETER, La $\sigma$-algèbre asymptotique d’une chaîne de Galton-Watson, Ann. Inst. H. Poincaré 13 (1977), no. 3, 193–230.
• [27] R. LYONS and Y. PERES, Probability on trees and networks., vol. 42, Cambridge: Cambridge University Press, 2016 (English).
• [28] J. NEVEU, Arbres et processus de Galton-Watson, Ann. de l’Inst. Henri Poincaré 22 (1986), 199–207.
• [29] J. NEVEU, Sur le théorème ergodique de Chung-Erdős, C. R. Acad. Sci. Paris 257 (1963), 2953–2955.
• [30] L. OVERBECK, Martin boundaries of some branching processes, Ann. Inst. H. Poincaré 30 (1994), no. 2, 181–195.
• [31] F. PAPANGELOU, A lemma on the Galton-Watson process and some of its consequences, Proc. Amer. Math. Soc. 19 (1968), 1469–1479.
• [32] U. RÖSLER, A fixed point theorem for distributions, Stochastic Process. Appl. 42 (1992), no. 2, 195–214.
• [33] E. SENETA, On recent theorems concerning the supercritical Galton-Watson process, Ann. Math. Statist. 39 (1968), 2098–2102.
• [34] E. SENETA, Functional equations and the Galton-Watson process, Advances in Appl. Probability 1 (1969), 1–42.
• [35] V. I. WACHTEL, D. E. DENISOV, and D. A. KORSHUNOV, Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case, Proceedings of the Steklov Institute of Mathematics 282 (2013), no. 1, 273–297.