Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Edgeworth expansions for profiles of lattice branching random walks

Rudolf Grübel and Zakhar Kabluchko

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Consider a branching random walk on $\mathbb{Z}$ in discrete time. Denote by $L_{n}(k)$ the number of particles at site $k\in\mathbb{Z}$ at time $n\in\mathbb{N}_{0}$. By the profile of the branching random walk (at time $n$) we mean the function $k\mapsto L_{n}(k)$. We establish the following asymptotic expansion of $L_{n}(k)$, as $n\to\infty$: \begin{equation*}\mathrm{e}^{-\varphi(0)n}L_{n}(k)=\frac{\mathrm{e}^{-\frac{1}{2}x_{n}^{2}(k)}}{\sqrt{2\pi\varphi"(0)n}}\sum_{j=0}^{r}\frac{F_{j}(x_{n}(k))}{n^{j/2}}+o(n^{-\frac{r+1}{2}})\quad \text{a.s.},\end{equation*} where $r\in\mathbb{N}_{0}$ is arbitrary, $\varphi(\beta)=\log\sum_{k\in\mathbb{Z}}\mathrm{e}^{\beta k}\mathbb{E}L_{1}(k)$ is the cumulant generating function of the intensity of the branching random walk and \begin{equation*}x_{n}(k)=\frac{k-\varphi'(0)n}{\sqrt{\varphi"(0)n}}.\end{equation*} The expansion is valid uniformly in $k\in\mathbb{Z}$ with probability $1$ and the $F_{j}$’s are polynomials whose random coefficients can be expressed through the derivatives of $\varphi$ and the derivatives of the limit of the Biggins martingale at $0$. Using exponential tilting, we also establish more general expansions covering the whole range of the branching random walk except its extreme values. As an application of this expansion for $r=0,1,2$ we recover in a unified way a number of known results and establish several new limit theorems. In particular, we study the a.s. behavior of the individual occupation numbers $L_{n}(k_{n})$, where $k_{n}\in\mathbb{Z}$ depends on $n$ in some regular way. We also prove a.s. limit theorems for the mode $\mathop{\operatorname{arg\,max}}_{k\in\mathbb{Z}}L_{n}(k)$ and the height $\max_{k\in\mathbb{Z}}L_{n}(k)$ of the profile. The asymptotic behavior of these quantities depends on whether the drift parameter $\varphi'(0)$ is integer, non-integer rational, or irrational. Applications of our results to profiles of random trees including binary search trees and random recursive trees will be given in a separate paper.


Nous considérons une marche branchante sur $\mathbb{Z}$ en temps discret. Soit $L_{n}(k)$ le nombre de particules au site $k\in\mathbb{Z}$ au temps $n\in\mathbb{N}_{0}$. Nous appelons profil de la marche branchante (au temps $n$) la fonction $k\mapsto L_{n}(k)$. Nous établissons le développement asymptotique suivant pour $L_{n}(k)$, lorsque $n\to\infty$ : \begin{equation*}\mathrm{e}^{-\varphi(0)n}L_{n}(k)=\frac{\mathrm{e}^{-\frac{1}{2}x_{n}^{2}(k)}}{\sqrt{2\pi\varphi"(0)n}}\sum_{j=0}^{r}\frac{F_{j}(x_{n}(k))}{n^{j/2}}+o(n^{-\frac{r+1}{2}})\quad \text{p.s.},\end{equation*} où $r\in\mathbb{N}_{0}$ est arbitraire, $\varphi(\beta)=\log\sum_{k\in\mathbb{Z}}\mathrm{e}^{\beta k}\mathbb{E}L_{1}(k)$ est la fonction génératrice des cumulants de l’intensité de la marche branchante, et \begin{equation*}x_{n}(k)=\frac{k-\varphi'(0)n}{\sqrt{\varphi"(0)n}}.\end{equation*} Le développement est valable uniformément en $k\in\mathbb{Z}$ avec probabilité $1$ et les $F_{j}$ sont des polynômes dont les coefficients aléatoires s’expriment à l’aide des dérivées de $\varphi$ et des dérivées de la limite de la martingale de Biggins en $0$. En utilisant une déformation exponentielle, nous établissons aussi des développements plus généraux qui couvrent tout le spectre de la marche branchante à l’exception des valeurs extrêmes. Comme application de ce développement pour $r=0,1,2$ nous retrouvons de façon unifiée plusieurs résultats connus et montrons de nouveaux théorèmes limite. En particulier, nous étudions le comportement p.s. des nombres d’occupation $L_{n}(k_{n})$, où $k_{n}\in\mathbb{Z}$ dépend de $n$ de façon régulière. Nous montrons aussi un théorème limite p.s. pour le mode $\mathop{\operatorname{arg\,max}}_{k\in\mathbb{Z}}L_{n}(k)$ et la hauteur $\max_{k\in\mathbb{Z}}L_{n}(k)$ du profil. Le comportement asymptotique de ces quantités dépend de si le paramètre de la dérive $\varphi'(0)$ est entier, rationnel, ou irrationnel. D’autres applications de nos résultats aux profils d’arbres aléatoires, incluant les arbres de recherche binaires et les arbres aléatoires récursifs, seront donnés dans un autre article.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 2103-2134.

Received: 30 March 2015
Revised: 12 June 2016
Accepted: 10 August 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F10: Large deviations 60F15: Strong theorems

Branching random walk Edgeworth expansion Central limit theorem Profile Biggins martingale Random analytic function Mod-$\varphi$-convergence Height Mode


Grübel, Rudolf; Kabluchko, Zakhar. Edgeworth expansions for profiles of lattice branching random walks. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 2103--2134. doi:10.1214/16-AIHP785.

Export citation


  • [1] S. Asmussen and N. Kaplan. Branching random walks. I. Stochastic Process. Appl. 4 (1976) 1–13.
  • [2] R. R. Bahadur and R. Ranga Rao. On deviations of the sample mean. Ann. Math. Stat. 31 (1960) 1015–1027.
  • [3] R. N. Bhattacharya and R. Ranga Rao. Normal Approximation and Asymptotic Expansions. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York–London–Sydney, 1976.
  • [4] J. D. Biggins. Chernoff’s theorem in the branching random walk. J. Appl. Probab. 14 (3) (1977) 630–636.
  • [5] J. D. Biggins. Martingale convergence in the branching random walk. J. Appl. Probab. 14 (1977) 25–37.
  • [6] J. D. Biggins. Growth rates in the branching random walk. Z. Wahrsch. Verw. Gebiete 48 (1) (1979) 17–34.
  • [7] J. D. Biggins. The central limit theorem for the supercritical branching random walk, and related results. Stochastic Process. Appl. 34 (2) (1990) 255–274.
  • [8] J. D. Biggins. Uniform convergence of martingales in the one-dimensional branching random walk. In Selected Proceedings of the Sheffield Symposium on Applied Probability 159–173. Sheffield, 1989. IMS Lecture Notes Monogr. Ser. 18. Inst. Math. Statist, Hayward, CA, 1991.
  • [9] J. D. Biggins. Uniform convergence of martingales in the branching random walk. Ann. Probab. 20 (1) (1992) 137–151.
  • [10] J. D. Biggins. Branching out. In Probability and Mathematical Genetics 113–134. London Math. Soc. Lecture Note Ser. 378. Cambridge Univ. Press, Cambridge, 2010.
  • [11] J. D. Biggins and D. R. Grey. A note on the growth of random trees. Statist. Probab. Lett. 32 (4) (1997) 339–342.
  • [12] D. Blackwell and J. L. Hodges. The probability in the extreme tail of a convolution. Ann. Math. Stat. 30 (1959) 1113–1120.
  • [13] B. Chauvin, M. Drmota and J. Jabbour-Hattab. The profile of binary search trees. Ann. Appl. Probab. 11 (4) (2001) 1042–1062.
  • [14] B. Chauvin, T. Klein, J.-F. Marckert and A. Rouault. Martingales and profile of binary search trees. Electron. J. Probab. 10 (12) (2005) 420–435.
  • [15] X. Chen. Exact convergence rates for the distribution of particles in branching random walks. Ann. Appl. Probab. 11 (4) (2001) 1242–1262.
  • [16] F. Delbaen, E. Kowalski and A. Nikeghbali. Mod-$\phi$ convergence. Int. Math. Res. Not. IMRN 11 (2015) 3445–3485.
  • [17] L. Devroye and H.-K. Hwang. Width and mode of the profile for some random trees of logarithmic height. Ann. Appl. Probab. 16 (2) (2006) 886–918.
  • [18] M. Drmota. Random Trees. An Interplay Between Combinatorics and Probability. Springer, Wien, 2009.
  • [19] M. Drmota and H.-K. Hwang. Profiles of random trees: Correlation and width of random recursive trees and binary search trees. Adv. in Appl. Probab. 37 (2) (2005) 321–341.
  • [20] M. Drmota, S. Janson and R. Neininger. A functional limit theorem for the profile of search trees. Ann. Appl. Probab. 18 (1) (2008) 288–333.
  • [21] V. Féray, P.-L. Méliot and A. Nikeghbali. Mod-$\phi$ convergence and precise deviations. Preprint, 2015. Available at
  • [22] M. Fuchs, H.-K. Hwang and R. Neininger. Profiles of random trees: Limit theorems for random recursive trees and binary search trees. Algorithmica 46 (3–4) (2006) 367–407.
  • [23] Z. Gao and Q. Liu Exact convergence rates in central limit theorems for a branching random walk with a random environment in time. Preprint, 2014. Available at
  • [24] Z. Gao, Q. Liu and H. Wang. Central limit theorems for a branching random walk with a random environment in time. Acta Math. Sci. Ser. B Engl. Ed. 34 (2) (2014) 501–512.
  • [25] R. Grübel and Z. Kabluchko. A functional central limit theorem for branching random walks, almost sure weak convergence, and applications to random trees. Ann. Appl. Probab. 26 (6) (2016) 3659–3698.
  • [26] P. Hall. The Bootstrap and Edgeworth Expansion. Springer-Verlag, New York, 1992.
  • [27] T. E. Harris. The Theory of Branching Processes. Die Grundlehren der Mathematischen Wissenschaften, Bd. 119. Springer-Verlag, Berlin, 1963.
  • [28] A. Joffe and A. R. Moncayo. Random variables, trees, and branching random walks. Adv. Math. 10 (1973) 401–416.
  • [29] Z. Kabluchko. Distribution of levels in high-dimensional random landscapes. Ann. Appl. Probab. 22 (1) (2012) 337–362.
  • [30] N. Kaplan and S. Asmussen. Branching random walks. II. Stochastic Process. Appl. 4 (1976) 15–31.
  • [31] Z. Katona. Width of a scale-free tree. J. Appl. Probab. 42 (3) (2005) 839–850.
  • [32] R. Neininger. Refined quicksort asymptotics. Random Structures Algorithms 46 (2) (2015) 346–361.
  • [33] V. V. Petrov. On the probabilities of large deviations for sums of independent random variables. Theory Probab. Appl. 10 (1965) 287–298.
  • [34] V. V. Petrov. Sums of Independent Random Variables. Ergebnisse der Mathematik und Ihrer Grenzgebiete 82. Springer-Verlag, New York–Heidelberg, 1975.
  • [35] P. Révész, J. Rosen and Z. Shi. Large-time asymptotics for the density of a branching Wiener process. J. Appl. Probab. 42 (4) (2005) 1081–1094.
  • [36] U. Rösler, V. Topchii and V. Vatutin. Convergence rate for stable weighted branching processes. In Mathematics and Computer Science II (Versailles, 2002), Trends Math. 441–453. Birkhäuser, Basel, 2002.
  • [37] A. J. Stam. On a conjecture by Harris. Z. Wahrsch. Verw. Gebiete 5 (1966) 202–206.
  • [38] H. Sulzbach. A functional limit law for the profile of plane-oriented recursive trees. In Fifth Colloquium on Mathematics and Computer Science 339–350. Discrete Math. Theor. Comput. Sci. Proc., AI. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2008.
  • [39] K. Uchiyama. Spatial growth of a branching process of particles living in $\mathbf{R}^{d}$. Ann. Probab. 10 (4) (1982) 896–918.
  • [40] N. Yoshida. Central limit theorem for branching random walks in random environment. Ann. Appl. Probab. 18 (4) (2008) 1619–1635.