Central limit theorem for bifurcating markov chains under L2-ergodic conditions

Bifurcating Markov chains (BMCs) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for additive functionals of BMCs under $L^2$ -ergodic conditions with three different regimes. This completes the pointwise approach developed in a previous work. As an application, we study the elementary case of a symmetric bifurcating autoregressive process, which justifies the nontrivial hypothesis considered on the kernel transition of the BMCs. We illustrate in this example the phase transition observed in the fluctuations.

Exponential Growth of Bifurcating Processes with Ancestral Dependence

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.

Exponential Growth of Bifurcating Processes with Ancestral Dependence

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.