We study the exponential growth of branching processes with ancestral dependence. We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. Lifetimes and new cells’s numbers are also assumed to be dependent. Applying the spectral study of Laplace-type operators recently made in Hervé et al. [ESAIM: PS 23 (2019) 607–637], we illustrate our results in the Markov context, for which the exponential growth property is linked to the Laplace transform of the lifetimes of the cells.
Keywords: Markov processes, quasi-compactness, operator, perturbation, ergodicity, Laplace transform, branching process, age-dependent process, Malthusian parameter
Hervé, Loïc 1 ; Louhichi, Sana 1 ; Pène, Françoise 1
@article{PS_2019__23__584_0,
author = {Herv\'e, Lo{\"\i}c and Louhichi, Sana and P\`ene, Fran\c{c}oise},
title = {Exponential growth of branching processes in a general context of lifetimes and birthtimes dependence},
journal = {ESAIM: Probability and Statistics},
pages = {584--606},
year = {2019},
publisher = {EDP Sciences},
volume = {23},
doi = {10.1051/ps/2019001},
mrnumber = {4011568},
zbl = {1506.60067},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2019001/}
}
TY - JOUR AU - Hervé, Loïc AU - Louhichi, Sana AU - Pène, Françoise TI - Exponential growth of branching processes in a general context of lifetimes and birthtimes dependence JO - ESAIM: Probability and Statistics PY - 2019 SP - 584 EP - 606 VL - 23 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ps/2019001/ DO - 10.1051/ps/2019001 LA - en ID - PS_2019__23__584_0 ER -
%0 Journal Article %A Hervé, Loïc %A Louhichi, Sana %A Pène, Françoise %T Exponential growth of branching processes in a general context of lifetimes and birthtimes dependence %J ESAIM: Probability and Statistics %D 2019 %P 584-606 %V 23 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/ps/2019001/ %R 10.1051/ps/2019001 %G en %F PS_2019__23__584_0
Hervé, Loïc; Louhichi, Sana; Pène, Françoise. Exponential growth of branching processes in a general context of lifetimes and birthtimes dependence. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 584-606. doi: 10.1051/ps/2019001
[1] , Positive transfer operators and decay of correlations. World Scientific Publishing Co. Inc., River Edge, NJ (2000) | MR | Zbl | DOI
[2] , Ancestral lineage and limit theorems for branching Markov chains. To appear in J. Theor. Probab. 32 (2019) 249–281 | MR | Zbl | DOI
[3] , , and , Limit theorems for Markov processes indexed by continuous time Galton-Watson trees. Ann. Appl. Probab. 21 (2011) 2263–2314 | MR | Zbl | DOI
[4] and , On the theory of age-dependent stochastic branching processes. Proc. Nat. Acad. Sci. USA 34 (1948) 601–604 | MR | Zbl | DOI
[5] and , Diffusion approximation of a Knudsen gas model: dependence of the diffusion constant upon the boundary condition. Asymptot. Anal. 31 (2002) 93–111 | MR | Zbl
[6] and , Detection of cellular aging in a Galton-Watson processes. Stoch. Process. Appl. 120 (2010) 2495–2519 | MR | Zbl | DOI
[7] and , Iterated random functions. SIAM Rev. 41 (1999) 45–76 | MR | Zbl | DOI
[8] , Random Iterative Models. Applications of Mathematics. Springer-Verlag, Berlin, Heidelberg (1997) | MR | Zbl
[9] and , Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. Henri Poincaré 24 (1988) 73–98 | MR | Zbl | Numdam
[10] and , On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks. Ergodic Theory Dyn. Syst. 28 (2008) 423–446 | MR | Zbl | DOI
[11] , Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Probab. 17 (2007) 1538–1569 | MR | Zbl | DOI
[12] , The theory of branching processes. Springer-Verlag, Berlin (1963) | MR | Zbl | DOI
[13] , Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Am. Math. Soc. 118 (1993) 627–634 | MR | Zbl
[14] and , The Nagaev-Guivarch́ method via the Keller-Liverani theorem. Bull. Soc. Math. France 138 (2010) 415–489 | MR | Zbl | Numdam | DOI
[15] , and , Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains. ESAIM: PS 23 (2019) 607–637 | MR | Numdam | Zbl | DOI
[16] and , Branching Markov processes and related asymptotics. J. Multivar. Anal. 100 (2009) 1155–1167 | MR | Zbl | DOI
[17] and , Stability of the Spectrum for Transfer Operators. Annali della Scuola Normale Superiore di Pisa – Classe di Scienze Sér. 4 28 (1999) 141–152 | MR | Zbl | Numdam
[18] and , Branching processes in Biology. Springer-Verlag, New York (2002) | MR | Zbl | DOI
[19] and , Spectral theory and limit theorems for geometrically Ergodic Markov processes. Ann. Appl. Probab. 13 (2003) 304–362 | MR | Zbl | DOI
[20] and , Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes. Electr. J. Probab. 10 (2005) 61–123 | MR | Zbl
[21] and , Exponential growth of bifurcating processes with ancestral dependence. Adv. Appl. Probab. 47 (2015) 545–564 | MR | Zbl | DOI
[22] , Some limit theorems for stationary Markov chains. Theory Probab. Appl. 11 (1957) 378–406 | MR | Zbl | DOI
[23] , More exact statements of limit theorems for homogeneous Markov chains. Theory Probab. Appl. 6 (1961) 62–81 | Zbl | MR | DOI
Cité par Sources :
