Journal of Theoretical Probability

, Volume 32, Issue 1, pp 249–281 | Cite as

Ancestral Lineages and Limit Theorems for Branching Markov Chains in Varying Environment

  • Vincent BansayeEmail author


We consider branching processes in discrete time for structured population in varying environment. Each individual has a trait which belongs to some general state space and both its reproduction law and the trait inherited by its offsprings may depend on its trait and the environment. We study the long-time behavior of the population and the ancestral lineage of typical individuals under general assumptions. We focus on the mean growth rate and the trait distribution among the population. The approach relies on many-to-one formulae and the analysis of the genealogy, and a key role is played by well-chosen (possibly non-homogeneous) Markov chains. The applications use large deviations principles and the Harris ergodicity for these auxiliary Markov chains.


Branching processes Markov chains Varying environment Genealogies 

Mathematics Subject Classification (2010)

60J80 60J05 60F05 60F10 



This work was partially funded by Chair Modelisation Mathematique et Biodiversite VEOLIA-Ecole Polytechnique-MNHN-F.X, the professorial chair Jean Marjoulet, the project MANEGE ‘Modèles Aléatoires en Écologie, Génétique et Évolution’ 09-BLAN-0215 of ANR (French National Research Agency). The author is also grateful to Clément Dombry for mentioning [48].


  1. 1.
    Athreya, K., Kang, H.J.: Some limit theorems for positive recurrent branching Markov chains I. Adv. Appl. Prob. 30(3), 693–710 (1998)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Athreya, K., Kang, H.J.: Some limit theorems for positive recurrent branching Markov chains II. Adv. Appl. Prob. 30(3), 711–722 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Athreya, K.B., Karlin, S.: On branching processes with random environments II: limit theorems. Ann. Math. Stat. 42, 1843–1858 (1971)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Athreya, K.B.: Change of measure of Markov chains and the \(L\log L\) theorem for branching processes. Bernoulli 6(2), 323–338 (2000)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Baillon, J.-B., Clément, P., Greven, A., den Hollander, F.A.: Variational approach to branching random walk in random environment. Ann. Probab. 21(1), 290–317 (1993)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bansaye, V.: Proliferating parasites in dividing cells: Kimmel’s branching model revisited. Ann. Appl. Probab. 18(3), 967–996 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bansaye, V., Delmas, J.-F., Marsalle, L., Tran, V.C.: Limit theorems for Markov processes indexed by continuous time Galton–Watson trees. Ann. Appl. Probab. 21(6), 2263–2314 (2011)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bansaye, V., Lambert, A.: New approaches of source-sink metapopulations decoupling the roles of demography and dispersal. To appear in Theor. Pop, Biology (2012)Google Scholar
  9. 9.
    Bansaye, V., Huang, C.: Weak law of large numbers for some Markov chains along non homogeneous genealogies. ESAIM 19, 307–326 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bansaye, V., Camanès, A.: Aging branching process and queuing for an infinite bus line. (2016). arXiv:1506.04168
  11. 11.
    Biggins, J.D.: Martingale convergence in the branching random walk. J. Appl. Probab. 14, 25–37 (1977)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Biggins, J.D., Cohn, H., Nerman, O.: Multi-type branching in varying environment. Stoch. Proc. Appl. 83, 357–400 (1999)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Biggins, J.D.: The central limit theorem for the supercritical branching random walk and related results. Stoch. Proc. Appl. 34, 255–274 (1990)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Cloez, B.: Limit theorems for some branching measure-valued processes. (2011). arXiv:1106.0660
  15. 15.
    Chauvin, B., Rouault, A.: KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees. Probab. Theory Relat. Fields 80(2), 299–314 (1988)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Chauvin, B., Rouault, A., Wakolbinger, A.: Growing conditioned trees. Stoch. Process. Appl. 39, 117–130 (1991)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Chow, Y., Teicher, H.: Probability Theory: Independence. Interchangeability, Martingales (1988)zbMATHGoogle Scholar
  18. 18.
    Cohn, H.: On the growth of the supercritical multitype branching processes in random environment. Ann. Probab. 17(3), 1118–1123 (1989)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Comets, F., Popov, S.: On multidimensional branching random walks in random environment. Ann. Probab. 35(1), 68–114 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Comets, F., Popov, S.: Shape and local growth for multidimensional branching random walks in random environment. ALEA 3, 273–299 (2007)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Comets, F., Yoshida, N.: Branching random walks in space-time random environment: survival probability, global and local growth rates. J. Theor. Probab. 24, 657–687 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Dembo, A., Zeitouni, O.: Large deviations techniques and applications. Applications of Mathematics (New York) 38, 2nd edn. (1998)Google Scholar
  23. 23.
    Delmas, J.-F., Marsalle, L.: Detection of cellular aging in a Galton–Watson process. Stoch. Process. Appl. 120, 2495–2519 (2010)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ethier, S.N., Kurtz, T.G.: Markov Processes. Characterization and Convergence. John Wiley & Sons, New York (1986)zbMATHGoogle Scholar
  25. 25.
    Gorostiza, L.G., Roelly, S., Wakolbinger, A.: Persistence of critical multitype particle and measure branching processes. Probab. Theory Relat. Fields 92(3), 313–335 (1992)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Kallenberg, O.: Stability of critical cluster fields. Math. Nachr. 77, 7–43 (1977)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Furstenberg, H., Kesten, H.: Products of random matrices. Ann. Math. Stat. 31(2), 457–469 (1960)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Gantert, N., Müller, S., Popov, S., Vachkovskaia, M.: Survival of branching random walks in random environment. J. Theor. Probab. 23, 1002–1014 (2010)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Geiger, J.: Elementary new proofs of classical limit theorems for Galton–Watson processes. J. Appl. Prob. 36, 301–309 (1999)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Georgii, H.O., Baake, E.: Supercritical multitype branching processes: the ancestral types of typical individuals. Adv. Appl. Probab. 35(4), 1090–1110 (2003)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Guyon, J.: Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Probab. 17, 1538–1569 (2007)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Harris, S. C., Roberts, M. I.: The many-to-few lemma and multiple spines. in Ann. Inst. (2012) Henri Poincaré, arXiv:1106.4761
  33. 33.
    Harris, S.C., Roberts, M.I.: A strong law of large numbers for branching processes: almost sure spine events. Electr. Commun. Probab. 19, 28 (2013)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Jagers, P., Nerman, O.: The asymptotic composition of supercritical multi-type branching populations. In: Séminaire de Probabilités, XXX, volume 1626 of Lecture Notes in Math., pages 40-54. Springer, Berlin (1996)Google Scholar
  35. 35.
    Jones, O.D.: On the convergence of multitype branching processes with varying environments. Ann. Appl. Probab. 7, 772–801 (1997)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Kaplan, N.: Some results about multidimensional branching processes with random environments. Ann. Probab. 2(3), 441–455 (1974)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Kurtz, T.: Inequalities for law of large numbers. Ann. Math. Statist. 43, 1874–1883 (1972)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Kurtz, T., Lyons, R., Pemantle, R., Peres, Y.: A conceptual proof of the Kesten–Stigum theorem for multi-type branching processes. In: Athreya, K.B., Jagers, P. (eds.) Classical and Modern Branching Processes, pp. 181–185. Springer, New York (1997)Google Scholar
  39. 39.
    Lyons, R., Pemantle, R., Peres, Y.: Conceptual proofs of \(L\log L\) criteria for mean behavior of branching processes. Ann. Probab. 23(3), 1125–1138 (1995)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Lyons, R.: A simple path to Biggins’ martingale convergence for branching random walk. In: Classical and Modern Branching Processes (Minneapolis, MN, 1994), 84 of IMA Vol. Math. Appl., pp. 217–221. Springer, New York (1997)Google Scholar
  41. 41.
    Meyn, S., Tweedie, L.: Markov Chains and Stochastic Stability. Springer, Berlin (2009)zbMATHGoogle Scholar
  42. 42.
    Mode, C.J.: Multi-Type Branching Processes-Theory and Application, p. 300. American Elsevier Publishing Company, Inc., New York (1971)Google Scholar
  43. 43.
    Mischler, S., Scher, J.: Spectral analysis of semigroups and growth-fragmentation equations To appear in Ann. Inst. H. Poincaré Anal. Non Linéaire (2013). arXiv:1310.7773
  44. 44.
    Mukhamedov, F.: Weak ergodicity of nonhomogeneous Markov chains on noncommutative L1-spaces. Banach J. Math. Anal. 7(2), 53–73 (2013)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Nakashima, M.: Almost sure central limit theorem for branching random walks in random environment. Ann. Appl. Probab. 21(1), 351–373 (2011)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Nerman, O., Jagers, P.: The stable double infinite pedigree process of supercritical branching populations. Z. Wahrsch. Verw. Gebiete 65(3), 445–460 (1984)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Seneta, E.: Non-negative Matrices and Markov Chains, p. 21. Springer, Berlin (2006)zbMATHGoogle Scholar
  48. 48.
    Seppäläinen, T.: Large Deviations for Markov chains with Random Transitions. Ann. Prob. 22(2), 713–748 (1994)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Tanny, D.: A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means. Stoch. Process. Appl. 28(1), 123–139 (1988)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Yoshida, N.: Central limit theorem for random walk in random environment. Ann. Appl. Probab. 18(4), 1619–1635 (2008)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.CMAP, Ecole Polytechnique, CNRSPalaiseau CedexFrance

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