Advances in Applied Probability

Exponential growth of bifurcating processes with ancestral dependence

Sana Louhichi and Bernard Ycart

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under the hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the independent and identically distributed supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of multiplicative ergodicity, in a similar way as to the Laplace transform of lifetimes in the i.i.d. case. An identifiable model for which the multiplicative ergodicity coefficients and the growth rate can be explicitly computed is proposed.

Article information

Source
Adv. in Appl. Probab., Volume 47, Number 2 (2015), 545-564.

Dates
First available in Project Euclid: 25 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1435236987

Digital Object Identifier
doi:10.1239/aap/1435236987

Mathematical Reviews number (MathSciNet)
MR3360389

Zentralblatt MATH identifier
1318.60088

Subjects
Primary: 60J85: Applications of branching processes [See also 92Dxx]
Secondary: 92D25: Population dynamics (general)

Keywords
Branching process bifurcating Markov chain multiplicative ergodicity cell kinetics

Citation

Louhichi, Sana; Ycart, Bernard. Exponential growth of bifurcating processes with ancestral dependence. Adv. in Appl. Probab. 47 (2015), no. 2, 545--564. doi:10.1239/aap/1435236987. https://projecteuclid.org/euclid.aap/1435236987


Export citation

References

  • Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • Bellman, R. and Harris, T. (1952). On age-dependent binary branching processes. Ann. Math. (2) 55, 280–295.
  • Benjamini, I. and Peres, Y. (1994). Markov chains indexed by trees. Ann. Prob. 22, 219–243.
  • Bitseki Penda, S. V., Djellout, H. and Guillin, A. (2014). Deviation inequelities, moderate deviations and some limit theorems for bifurcating Markov chains with application. Ann. Appl. Prob. 24, 235–291.
  • Cowan, R. and Staudte, R. (1986). The bifurcating autoregression model in cell lineage studies. Biometrics 42, 769–783.
  • Crump, K. S. and Mode, C. J. (1969). An age-dependent branching process with correlations among sister cells. J. Appl. Prob. 6, 205–210.
  • De la Peña, V. H. and Lai, T. L. (2001). Theory and applications of decoupling. In Probability and Statistical Models with Applications, Chapman & Hall/CRC, Boca Raton, FL, pp. 117–145.
  • De Saporta, B., Gégout-Petit, A. and Marsalle, L. (2011). Parameters estimation for asymmetric bifurcating autoregressive processes with missing data. Electron. J. Statist. 5, 1313–1353.
  • Delmas, J.-F. and Marsalle, L. (2010). Detection of cellular aging in a Galton–Watson process. Stoch. Process. Appl. 120, 2495–2519.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.
  • Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.
  • Glynn, P. W. and Whitt, W. (1994). Large deviations behavior of counting processes and their inverses. Queueing Systems Theory Appl. 17, 107–128.
  • Guyon, J. (2007). Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Prob. 17, 1538–1569.
  • Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.
  • Harvey, J. D. (1972). Synchronous growth of cells and the generation time distribution. J. General Microbiol. 70, 99–107.
  • Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables, with applications. Ann. Statist. 11, 286–295.
  • John, P. C. L. (ed.) (1981). The Cell Cycle. Cambridge University Press.
  • Kendall, D. G. (1952). On the choice of a mathematical model to represent normal bacterial growth. J. R. Statist. Soc. B 14, 41–44.
  • Kleptsyna, M. L., Le Breton, A. and Viot, M. (2002). New formulas concerning Laplace transforms of quadratic forms for general Gaussian sequences. J. Appl. Math. Stoch. Anal. 15, 323–339.
  • Kleptsyna, M., Le Breton, A. and Ycart, B. (2014). Exponential transform of quadratic functional and multiplicative ergodicity of a Gauss–Markov process. Statist. Prob. Lett. 87, 70–75.
  • Kontoyiannis, I. and Meyn, S. P. (2003). Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Prob. 13, 304–362.
  • Korevaar, J. (2004). Tauberian Theory. A Century of Developments. Springer, Berlin.
  • Markham, J. F. et al. (2010). A minimum of two distinct heritable factors are required to explain correlation structures in proliferating lymphocytes. J. R. Soc. Interface 7, 1049–1059.
  • Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.
  • Nordon, R. E., Ko, K.-H., Odell, R. and Schroeder, T. (2011). Multi-type branching models to describe cell differentiation programs. J. Theoret. Biol. 277, 7–18.
  • Pemantle, R. (1992). Automorphism invariant measures on trees. Ann. Prob. 20, 1549–1566.
  • Pemantle, R. (1995). Tree-indexed processes. Statist. Sci. 10, 200–213.
  • Pitt, M. K., Chatfield, C. and Walker, S. G. (2002). Constructing first order stationary autoregressive models via latent processes. Scand. J. Statist. 29, 657–663.
  • Powell, E. O. (1956). Growth rate and generation time of bacteria with special reference to continuous culture. Microbiol. 15, 492–511.
  • Rahn, O. (1932). A chemical explanation of the variability of the growth rate. J. Gen. Physiol. 15, 257–277.
  • Shao, Q.-M. (2000). A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theoret. Prob. 13, 343–356.
  • Spitzer, F. (1975). Markov random fields on an infinite tree. Ann. Prob. 3, 387–398.
  • Wang, P. et al. (2010). Robust growth of Escherichia coli. Curr. Biol. 20, 1099–1103.