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Quasi-stationary distributions for subcritical branching Markov chains

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  10 September 2025

Wenming Hong  and
Dan Yao Open the ORCID record for Dan Yao [Opens in a new window]
Wenming Hong*
Affiliation:
Beijing Normal University
Dan Yao*
Affiliation:
Beijing Normal University
*
*Postal address: School of Mathematical Sciences and Laboratory of Mathematics and Complex Systems, Beijing 100875, P.R. China.
*Postal address: School of Mathematical Sciences and Laboratory of Mathematics and Complex Systems, Beijing 100875, P.R. China.
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Abstract

Consider a subcritical branching Markov chain. Let $Z_n$ denote the counting measure of particles of generation n. Under some conditions, we give a probabilistic proof for the existence of the Yaglom limit of $(Z_n)_{n\in\mathbb{N}}$ by the moment method, based on the spinal decomposition and the many-to-few formula. As a result, we give explicit integral representations of all quasi-stationary distributions of $(Z_n)_{n\in\mathbb{N}}$, whose proofs are direct and probabilistic, and do not rely on Martin boundary theory.

Keywords

Branching Markov chainYaglom limitquasi-stationary distributionconditional reduced Galton–Watson process

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems
Secondary: 60F10: Large deviations

Information

Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 47
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Adékon, E. (2013). Convergence in law of the minimum of a branching random walk. Ann. Probab. 41, 1362–1426.Google Scholar
Asmussen, S. and Hering, H. (1983). Branching processes. In Progress in Probability and Statistics, Vol. 3. Birkhäuser Boston, Inc., Boston, MA.Google Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching processes. In Die Grundlehren der mathematischen Wissenschaften, Vol. Band 196. Springer-Verlag, New York-Heidelberg.10.1007/978-3-642-65371-1CrossRefGoogle Scholar
Bansaye, V. (2019). Ancestral lineages and limit theorems for branching Markov chains in varying environment. J. Theoret. Probab. 32, 249–281.10.1007/s10959-018-0825-1CrossRefGoogle Scholar
Bansaye, V. and Huang, C. (2015). Weak law of large numbers for some Markov chains along non homogeneous genealogies. ESAIM Probab. Stat. 19, 307–326.10.1051/ps/2014027CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I. Elementary Theory and Methods, 2nd edn. Probability and its Applications (New York). Springer-Verlag, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II. General Theory and Structure, 2nd edn. Probability and its Applications (New York). Springer, New York.Google Scholar
Durrett, R. (2010). Probability: Theory and Examples, 4nd edn, Vol. 31 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge.Google Scholar
Evans, S. N. and Perkins, E. (1990). Measure-valued Markov branching processes conditioned on nonextinction. Israel J. Math. 71, 329–337.10.1007/BF02773751CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. I, 3nd edn. John Wiley & Sons, Inc., New York-London-Sydney.Google Scholar
Fleischmann, K. and Prehn, U. (1974). Ein Grenzwertsatz f $\ddot{u}$ r subkritische Verzweigungsprozesse mit endlich vielen Typen von Teilchen. Math. Nachr. 64, 357–362.10.1002/mana.19740640123CrossRefGoogle Scholar
Fleischmann, K. and Siegmund-Schultze, R. (1977). The structure of reduced critical Galton–Watson processes. Math. Nachr. 79, 233–241.10.1002/mana.19770790121CrossRefGoogle Scholar
Fleischmann, K. and Siegmund-Schultze, R. (1978). An invariance principle for reduced family trees of critical spatially homogeneous branching processes. Serdica 4. 2-3, 111–134.Google Scholar
Fleischmann, K. and Vatutin, V. A. (1999). Reduced subcritical Galton–Watson processes in a random environment. Adv. Appl. Probab. 31, 88–111.10.1239/aap/1029954268CrossRefGoogle Scholar
Folland, G. (1999). Real Analysis: Modern Techniques and Their Applications, 2nd edn. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York.Google Scholar
Geiger, J. (1999). Elementary new proofs of classical limit theorems for Galton–Watson processes. J. Appl. Probab. 36, 301–309.10.1239/jap/1032374454CrossRefGoogle Scholar
Grafakos, L. (2014). Classical Fourier Analysis, 3nd edn. Springer, New York.10.1007/978-1-4939-1194-3CrossRefGoogle Scholar
Harris, S. C., Horton, E., Kyprianou, A. E. and Wang, M. (2022). Yaglom limit for critical nonlocal branching Markov processes. Ann. Probab. 50, 2373–2408.10.1214/22-AOP1585CrossRefGoogle Scholar
Harris, S. C. and Roberts, M. I. (2017). The many-to-few lemma and multiple spines. Ann. Inst. Henri Poincaré Probab. Stat. 53, 226–242.10.1214/15-AIHP714CrossRefGoogle Scholar
Heathcote, C. R., Seneta, E. and Vere-Jones, D. (1967). A refinement of two theorems in the theory of branching processes. Teor. Verojatnost. i Primenen. 12, 341–346.Google Scholar
Hering, H. (1977). Minimal moment conditions in the limit theory for general Markov branching processes. Ann. Inst. H. Poincaré Sect. B (N.S.) 13, 299–319.Google Scholar
Hong, W. and Liang, S. (2024). Conditional central limit theorem for critical branching random walk. ALEA, Lat. Am. J. Probab. Math. Stat. 21, 555–574.10.30757/ALEA.v21-22CrossRefGoogle Scholar
Hong, W. and Yao, D. (2023). Conditional central limit theorem for subcritical branching random walk. ALEA, Lat. Am. J. Probab. Math. Stat. 20, 1411–1432.10.30757/ALEA.v20-53CrossRefGoogle Scholar
Hoppe, F. M. (1980). On a Schröder equation arising in branching processes. Aequationes Math. 20, 33–37.10.1007/BF02190491CrossRefGoogle Scholar
Hoppe, F. M. and Seneta, E. (1978). Analytical methods for discrete branching processes. Adv. Probab. Related Topics. 5, 219–261.Google Scholar
Hu, Y. and Shi, Z. (2009). Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37, 742–789.10.1214/08-AOP419CrossRefGoogle Scholar
Joffe, A. (1967). On the Galton-Waston branching process with mean less than one. Ann. Math. Statist. 38, 264–266.10.1214/aoms/1177699079CrossRefGoogle Scholar
Kallenberg, O. (2017). Random measures, theory and applications. In Probability Theory and Stochastic Modelling, Vol. 77. Springer, Cham.Google Scholar
Lambert, A. (2007). Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12, 420–446.10.1214/EJP.v12-402CrossRefGoogle Scholar
Lebedev, N. N. (1972). Special Functions and Their Applications. Dover Publications, Inc., New York.Google Scholar
Li, Z. (2011). Measure-valued branching Markov processes. In Probability and its Applications (New York). Springer, Heidelberg.10.1007/978-3-642-15004-3_2CrossRefGoogle Scholar
Sh, Liptser, R. and Shiryayev, A. N. (1989). Theory of Martingales. Translated from Russian by K. Dzjaparidze, Vol. 49 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht.Google Scholar
Liu, R., Ren, Y. X., Song, R. and Sun, Z. (2021). Quasi-stationary distributions for subcritical superprocesses. Stochastic Process. Appl. 132, 108–134.10.1016/j.spa.2020.10.007CrossRefGoogle Scholar
Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23, 1125–1138.Google Scholar
Maillard, P. (2018). The $\lambda$ -invariant measures of subcritical Bienaymé-Galton–Watson processes. Bernoulli 24, 297–315.Google Scholar
Méléard, S. and Villemonais, D. (2012). Quasi-stationary distributions and population processes. Probab. Surv. 9, 340–410.10.1214/11-PS191CrossRefGoogle Scholar
Rapenne, V. (2023). Invariant measures of critical branching random walks in high dimension. Electron. J. Probab. 28, 1–38.10.1214/23-EJP906CrossRefGoogle Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes, Vol. 4 of Applied Probability. Spring, New York.10.1007/978-0-387-75953-1CrossRefGoogle Scholar
Shi, Z. (2015). Branching Random Walks, Vol. 2151 of Lecture Notes in Mathematics. Springer, Cham.Google Scholar
Shiryaev, A. N. (1996). Probability, (English summary). Translated from the first (1980) Russian edition by R. P. Boas. 2nd edn., Vol. 95 of Graduate Texts in Mathematics. Springer-Verlag, New York.Google Scholar
Spitzer, F. (1967). Two explicit Martin boundary constructions. In Symposium on Probability Methods in Analysis (Loutraki, 1966), Vol. 31 of Lecture Notes in Mathematics. Springer, Berlin-New York, pp. 296–298.10.1007/BFb0061127CrossRefGoogle Scholar
Yaglom, A. M. (1947). Certain limit theorems of the theory of branching random processes. Dokl. Akad. Nauk SSSR(N.S.) 56, 795–798.Google Scholar