A Phase Transition for Large Values of Bifurcating Autoregressive Models
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A Phase Transition for Large Values of Bifurcating Autoregressive Models Vincent Bansaye1 · S. Valère Bitseki Penda2 Received: 27 January 2020 / Revised: 3 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We describe the asymptotic behavior of the number Z n [an , ∞) of individuals with a large value in a stable bifurcating autoregressive process, where an → ∞. The study of the associated first moment is equivalent to the annealed large deviation problem of an autoregressive process in a random environment. The trajectorial behavior of Z n [an , ∞) is obtained by the study of the ancestral paths corresponding to the large deviation event together with the environment of the process. This study of large deviations of autoregressive processes in random environment is of independent interest and achieved first. The estimates for bifurcating autoregressive process involve then a law of large numbers for non-homogenous trees. Two regimes appear in the stable case, depending on whether one of the autoregressive parameters is greater than 1 or not. It yields different asymptotic behaviors for large local densities and maximal value of the bifurcating autoregressive process. Keywords Branching process · Autoregressive process · Random environment · Large deviations Mathematics Subject Classification (2010) 60J80 · 60J20 · 60K37 · 60F10 · 60J20 · 60C05 · 92D25
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S. Valère Bitseki Penda [email protected] Vincent Bansaye [email protected]
1
CMAP, École Polytechnique, Route de Saclay, 91128 Palaiseau, France
2
Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Université de Bourgogne Franche-Comté, 21000 Dijon, France
123
Journal of Theoretical Probability
1 Introduction The bifurcating autoregressive (BAR) process X = (X n )n≥1 is a model for affine random transmission of a real value along a binary tree. To define this process, we consider a real value 1 , independent of a sequence of i.i.d bivariate random variable X random variables (η2k , η2k+1 ), k ≥ 1 with law N2 (0, Γ ), where
1ρ Γ = , ρ ∈ (−1, 1). ρ 1 Then, X is defined inductively for k ≥ 1 by
X 2k = α X k + η2k X 2k+1 = β X k + η2k+1 ,
(1)
where α, β are nonnegative real numbers. Informally, the value X k of individual k is randomly transmitted to its two offsprings 2k and 2k + 1 following an autoregressive process. The bifurcating autoregressive provides a simple model of random transmission along a binary tree. This model has been introduced in the symmetric case α = β by Cowan [19] and Cowan and Staudte [20] to analyze cell lineages data. It allows to study the evolution and the transmission of a trait after division, in particular it size or it growth rate. In these works, Cowan and Staudte have focused on the study of the bacteria Escherichia Coli. E. Coli is a rod-shaped bacterium which reproduces by dividing in the middle, producing two cells. Several extensions of their model have been proposed and we refer, e.g., to the works of Basawa and Higgins
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