Abstract
We consider a branching Wiener process in ℝd, in which particles reproduce as a super-critical Galton-Watson process and disperse according to a Wiener process. For B ⊂ ℝd, let Zn(B) be the number of particles of generation n located in B. The study of the central limit theorem and related results about the counting measure Zn(·) is important because such results give good descriptions of the configuration of the branching Wiener process at time n. In earlier works, the exact convergence rate in the central limit theorem and the asymptotic expansion until the third order have been given. Here, we establish the asymptotic expansion of any order in the central limit theorem under a moment condition of the form EX (log X)1+λ < ∞.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Asmussen S. Convergence rates for branching processes. Ann Probab, 1976, 4: 139–146
Asmussen S, Hering H. Branching Processes. Boston: Birkhäuser, 1983
Asmussen S, Kaplan N. Branching random walks. I. Stochastic Process Appl, 1976, 4: 1–13
Athreya K B, Ney P E. Branching Processes. New York: Springer-Verlag, 1972
Beals R, Wong R. Special Functions. Cambridge: Cambridge University Press, 2010
Biggins J D. The central limit theorem for the supercritical branching random walk, and related results. Stochastic Process Appl, 1990, 34: 255–274
Bingham N H, Doney R A. Asymptotic properties of supercritical branching processes. I: The Galton-Watson process. Adv Appl Probab, 1974, 6: 711–731
Chen X. Exact convergence rates for the distribution of particles in branching random walks. Ann Appl Probab, 2001, 11: 1242–1262
Chen X, He H. On large deviation probabilities for empirical distribution of supercritical branching random walks with unbounded displacements. Probab Theory Related Fields, 2019, 175: 255–307
Erdélyi A. Higher Transcendental Functions, Volume II. New York: McGraw-Hill Book Company, 1955
Gao Z Q. Exact convergence rate of the local limit theorem for branching random walks on the integer lattice. Stochastic Process Appl, 2017, 127: 1282–1296
Gao Z Q, Liu Q. Exact convergence rate in the central limit theorem for a branching random walk with a random environment in time. Stochastic Process Appl, 2016, 126: 2634–2664
Gao Z Q, Liu Q. First- and second-order expansions in the central limit theorem for a branching random walk. C R Math Acad Sci Paris, 2016, 354: 532–537
Gao Z Q, Liu Q. Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time. Bernoulli, 2018, 24: 772–800
Gao Z Q, Liu Q, Wang H. Central limit theorems for a branching random walk with a random environment in time. Acta Math Sci Ser B Engl Ed, 2014, 34: 501–512
Grübel R, Kabluchko Z. A functional central limit theorem for branching random walks, almost sure weak convergence and applications to random trees. Ann Appl Probab, 2016, 26: 3659–3698
Grübel R, Kabluchko Z. Edgeworth expansions for profiles of lattice branching random walks. Ann Inst H Poincaré Probab Statist, 2017, 53: 2103–2134
Harris T E. The Theory of Branching Processes. Berlin: Springer-Verlag, 1963
Hong W, Zhang X. Asymptotic behaviour of heavy-tailed branching processes in random environments. Electron J Probab, 2019, 24: 56, 17pp
Iksanov A, Kabluchko Z. A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk. J Appl Probab, 2016, 53: 1178–1192
Iksanov A, Liang X, Liu Q. On Lp-convergence of the Biggins martingale with complex parameter. J Math Anal Appl, 2019, 479: 1653–1669
Joffe A, Moncayo A R. Random variables, trees, and branching random walks. Adv Math, 1973, 10: 401–416
Kabluchko Z. Distribution of levels in high-dimensional random landscapes. Ann Appl Probab, 2012, 22: 337–362
Kaplan N, Asmussen S. Branching random walks. II. Stochastic Process Appl, 1976, 4: 15–31
Klebaner C F. Branching random walk in varying environments. Adv Appl Probab, 1982, 14: 359–367
Liang X, Liu Q. Regular variation of fixed points of the smoothing transform. Stochastic Process Appl, 2020, 130: 4104–4140
Nakashima M. Almost sure central limit theorem for branching random walks in random environment. Ann Appl Probab, 2011, 21: 351–373
Révész P. Random Walks of Infinitely Many Particles. River Edge: World Scientific, 1994
Révész P, Rosen J, Shi Z. Large-time asymptotics for the density of a branching Wiener process. J Appl Probab, 2005, 42: 1081–1094
Shiryaev A N. Probability, 2nd ed. New York: Springer-Verlag, 1996
Stam A J. On a conjecture by Harris. Z Wahrscheinlichkeitstheorie verw Gebiete, 1966, 5: 202–206
Wang Y, Liu Z, Liu Q, et al. Asymptotic properties of a branching random walk with a random environment in time. Acta Math Sci Ser B Engl Ed, 2019, 39: 1345–1362
Yoshida N. Central limit theorem for branching random walks in random environment. Ann Appl Probab, 2008, 18: 1619–1635
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11971063 and 11731012), Hunan Natural Science Foundation of China (Grant No. 2017JJ2271), the Natural Science Foundation of Guangdong Province of China (Grant No. 2015A030313628), and the French Government “Investissements d’Avenir” Program (Grant No. ANR-11-LABX-0020-01). The authors thank the anonymous referees for helpful comments and suggestions. The work has benefited from some visits of Quansheng Liu to the School of Mathematical Sciences of Beijing Normal University, whose financial support and hospitality have been well appreciated.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gao, ZQ., Liu, Q. Asymptotic expansions in the central limit theorem for a branching Wiener process. Sci. China Math. 64, 2759–2774 (2021). https://doi.org/10.1007/s11425-020-1776-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-020-1776-4