Section: New Results

Random coefficient bifurcating autoregressive processes

Participants : Benoîte de Saporta, Anne Gégout-Petit.

In the 80's, Cowan and Staudte [72] introduced Bifurcating Autoregressive processes (BAR) as a parametric model to study cell lineage data. A quantitative characteristic of the cells (e.g. growth rate, age at division) is recorded over several generations descended from an initial cell, keeping track of the genealogy to study inherited effects. As a cell usually gives birth to two offspring by division, such genealogies are naturally structured as binary trees. BAR processes are thus a generalization of autoregressive processes (AR) to this binary tree structure, by modeling each line of descent as a first order AR process, allowing the environmental effects on sister cells to be correlated. Statistical inference for the parameters of BAR processes has been widely studied, either based on the observation of a single tree growing to infinity [72] , [85] , [83] , [95] or on a large number of small independent trees [86] , [84] .

Various extensions of the original model have been proposed, but to our best knowledge, only two papers [71] and [70] deal with random coefficient BAR processes. In the former by Bui and Huggins it is explained that random coefficients BAR processes can account for observations that do not fit the usual BAR model. For instance, the extra randomness can model irregularities in nutrient concentrations in the media in which the cells are grown. In this work, we propose a new model for random coefficient BAR processes (R-BAR). It is more general than that of Bui and Huggins, as the random variables are not supposed to be Gaussian, they may not have moments of all order and correlation between all the sources of randomness are allowed. Moreover, we propose an asymmetric model in the continuance of [82] , [69] , [74] , [70] , [7] , [24] in the context of missing data. Indeed, experimental data are often incomplete and it is important to take this phenomenon into account for the inference. We model the structure of available data by a Galton Watson tree, instead of a complete binary tree. Our model is close to that developed in [70] , but the assumptions on the noise process are different as we allow correlation between the two sources of randomness but require higher moments because of the missing data and because we do not use a weighted estimator. The main difference is that the model in [70] is fully observed, whereas ours allows for missing observations.

Our approach for the inference of our model is also different from [71] , [70] . As we cannot use maximum likelihood estimation, we propose modified least squares estimators as in [91] . The originality of our approach is that it combines the bifurcating Markov chain and martingale approaches. Bifurcating Markov chains (BMC) were introduced in [82] on complete binary trees and further developed in [74] in the context of missing data on Galton Watson trees. BAR models can be seen as a special case of BMC. This interpretation allows us to establish the convergence of our estimators. A by-product of our procedure is a new general result for BMC on Galton Watson trees. Indeed, in [82] , [74] the driven noise sequence is assumed to have moments of all order. Here, we establish new laws of large numbers for polynomial functions of the BMC where the noise sequence only has moments up to a given order. The strong law of large numbers [78] and the central limit theorem for martingales have been previously used in the context of BAR processes and adapted to special cases of martingales on binary trees. In this paper, we establish a general law of large numbers for square integrable martingales on Galton Watson binary trees. This result is applied to our R-BAR model to obtain sharp convergence rates and a quadratic strong law for our estimators.

This work is in collaboration with Laurence Marsalle (Lille 1 University). It is submitted for publication [64] .