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A Phase Transition for Large Values of Bifurcating Autoregressive Models

Author

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  • Vincent Bansaye

    (CMAP, École Polytechnique)

  • S. Valère Bitseki Penda

    (Université de Bourgogne Franche-Comté)

Abstract

We describe the asymptotic behavior of the number $$Z_n[a_n,\infty )$$ Z n [ a n , ∞ ) of individuals with a large value in a stable bifurcating autoregressive process, where $$a_n\rightarrow \infty $$ a n → ∞ . 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 $$Z_n[a_n,\infty )$$ Z n [ a n , ∞ ) 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.

Suggested Citation

  • Vincent Bansaye & S. Valère Bitseki Penda, 2021. "A Phase Transition for Large Values of Bifurcating Autoregressive Models," Journal of Theoretical Probability, Springer, vol. 34(4), pages 2081-2116, December.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:4:d:10.1007_s10959-020-01033-w
    DOI: 10.1007/s10959-020-01033-w
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    References listed on IDEAS

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    1. Bercu, Bernard & Blandin, Vassili, 2015. "A Rademacher–Menchov approach for random coefficient bifurcating autoregressive processes," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1218-1243.
    2. S. Valère Bitseki Penda & Adélaïde Olivier, 2017. "Autoregressive functions estimation in nonlinear bifurcating autoregressive models," Statistical Inference for Stochastic Processes, Springer, vol. 20(2), pages 179-210, July.
    3. de Saporta, Benoîte & Gégout-Petit, Anne & Marsalle, Laurence, 2014. "Statistical study of asymmetry in cell lineage data," Computational Statistics & Data Analysis, Elsevier, vol. 69(C), pages 15-39.
    4. de Saporta, Benoîte & Gégout-Petit, Anne & Marsalle, Laurence, 2012. "Asymmetry tests for bifurcating auto-regressive processes with missing data," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1439-1444.
    5. J. Zhou & I. V. Basawa, 2005. "Maximum Likelihood Estimation for a First‐Order Bifurcating Autoregressive Process with Exponential Errors," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(6), pages 825-842, November.
    6. Delmas, Jean-François & Marsalle, Laurence, 2010. "Detection of cellular aging in a Galton-Watson process," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2495-2519, December.
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