Weak law of large numbers for some Markov chains along non homogeneous genealogies

Bansaye, Vincent; Huang, Chunmao

ESAIM: Probability and Statistics, Tome 19 (2015), pp. 307-326.

We consider a population with non-overlapping generations, whose size goes to infinity. It is described by a discrete genealogy which may be time non-homogeneous and we pay special attention to branching trees in varying environments. A Markov chain models the dynamic of the trait of each individual along this genealogy and may also be time non-homogeneous. Such models are motivated by transmission processes in the cell division, reproduction-dispersion dynamics or sampling problems in evolution. We want to determine the evolution of the distribution of the traits among the population, namely the asymptotic behavior of the proportion of individuals with a given trait. We prove some quenched laws of large numbers which rely on the ergodicity of an auxiliary process. A central limit is also established in the transient case.

S. Alili, Asymptotic behaviour for random walk in random environments. J. Appl. Probab. 36 (1999) 334–349. | MR | Zbl

K.B. Athreya and S. Karlin, On branching processes with random environments, I: extinction probability. Ann. Math. Stat. 42 (1971) 1499–1520. | MR | Zbl

K.B. Athreya and S. Karlin, On branching processes with random environments, II: limit theorems. Ann. Math. Stat. 42 (1971) 1843–1858. | MR | Zbl

K.B. Athreya and H.J. Kang, Some limit theorems for positive recurrent branching Markov chains I, II. Adv. Appl. Probab. 30 (1998) 693–722. | MR | Zbl

V. Bansaye, Proliferating parasites in dividing cells: Kimmel’s branching model revisited. Ann. Appl. Probab. 18 (2008) 967–996. | MR | Zbl

V. Bansaye and V.C. Tran, Branching feller diffusion for cell division with parasite infection. ALEA 8 (2011) 95–127. | MR | Zbl

V. Bansaye, J-F. Delmas, L. Marsalle and V.C. Tran, Limit theorems for Markov processes indexed by supercritical Galton Watson tree. Ann. Appl. Probab. 21 (2011) 2263–2314. | MR | Zbl

B. Bercu, B. De Saporta and A. Gégout-Petit, Asymptotic analysis for bifurcating autoregressive processes via a martingale approach. Electr. J. Probab. 14 (2009) 2492–2526. | MR | Zbl

J.D. Biggins, Martingale convergence in the branching random walk. J. Appl. Probab. 14 (1977) 25–37. | MR | Zbl

J.D. Biggins, Chernoff’s theorem in the branching random walk. J. Appl. Probab. 14 (1977) 630–636. | MR | Zbl

J.D. Biggins, The central limit theorem for the supercritical branching random walk, and related results. Stoch. Proc. Appl. 34 (1990) 255–274. | MR | Zbl

F. Comets and S. Popov, Shape and local growth for multidimensional branching random walks in random environment. ALEA 3 (2007) 273–299. | MR | Zbl

F. Comets and N. Yoshida, Branching random walks in space-time random environment: survival probability, global and local growth rates. J. Theor. Probab. 24 (2011) 657–687. | MR | Zbl

J-F. Delmas and L. Marsalle, Detection of cellular aging in a Galton-Watson process. Stoch. Proc. Appl. 120 (2010) 2495–2519. | MR | Zbl

M. Doumic, M. Hoffmann, N. Krell and L. Robert, Statistical estimation of a growth-fragmentation model observed on a genealogical tree. Bernoulli 21 (2015) 1760–1799. | MR | Zbl

H. Fréville, B. Colas, M. Riba, H. Caswell, A. Mignot, E. Imbert and I. Olivieri, Spatial and demographic variability in the endemic plant species Centaurea Corymbosa (Asteraceae). Ecology 85 (2004) 694–703.

L. Gallardo, A central limit theorem for Markov chains and applications to hypergroups. Proc. Am. Math. Soc. 127 (1999) 1837–1845. | MR | Zbl

N. Gantert, S. Müller, S. Popov and M. Vachkovskaia, Survival of branching random walks in random environment. J. Theor. Probab. 23 (2010) 1002–1014. | MR | Zbl

J. Guyon, Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Probab. 17 (2007) 1538–1569. | MR | Zbl

S.C. Harris and M. Roberts, The many-to-few lemma and multiple spines. Preprint available via (2014). | arXiv

C. Huang, Q. Liu, Convergence in $L^p$ and its exponential rate for a branching process in a random environment. Electron. J. Probab. 19 (2014) 1–22. | Zbl

C. Huang and Q. Liu, Branching random walk with a random environment in time. Preprint available via (2014). | arXiv | Zbl

N. Lartillot and R. Poujol, A phylogenetic model for investigating correlated evolution of substitution rates and continuous phenotypic characters. Mol. Biol. Evol. 28 (2010) 729–744.

Q. Liu, Branching random walks in random environment, in Proc. of the 4th International Congress of Chinese Mathematicians. Edited by L. Ji, K. Liu, L. Yang and S.-T. Yau, Vol. II (2007) 702–719.

S.P. Meyn and R.L. Tweedie, Stability of Markovian processes I: criteria for discrete-time chains. Adv. Appl. Probab. 24 (1992) 542–574. | Zbl

F. Mukhamedov, On $L_1$-weak ergodicity of nonhomogeneous discrete Markov processes and its applications. Available via (2012). | arXiv

S. Müller, A criterion for transience of multidimensional branching random walk in random environment. Electr. J. Probab. 13 (2008) 1189–1202. | Zbl

M. Nakashima, Almost sure central limit theorem for branching random walks in random environment. Ann. Appl. Probab. 21 (2011) 351–373. | Zbl

S. Orey, Markov chains with stationary Markov transition probabilities. Ann. Probab. 19 (1991) 907–928. | Zbl

H. Pan, Law of large number for even-odd Markov chain fields and a three times circulation Markov chain indexed by a tree. Ph.D. thesis (2010). http://globethesis.com/?t=2120360302966543.

T. Seppäläinen, Large deviations for Markov chains with random transitions. Ann. Probab. 22 (1994) 713–748. | Zbl

D.-A. Sinclair and L. Guarente, Extrachromosomal rDNA circles– a cause of aging in yeast. Cell 91 (1997) 1033–1042.

T. Stadler, Inferring speciation and extinction processes from extant species data. Proc. Natl. Acad. Sci. 108 (2011) 16145–16146.

E.J. Stewart, R. Madden, G. Paul and F. Taddei, Aging and death in a organism that reproduces by morphologically symmetric division. PLoS Biol. 3 (2005) e45.

N. Yoshida, Central limit theorem for random walk in random environment. Ann. Appl. Probab. 18 (2008) 1619–1635. | Zbl