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Central Limit Theorem for Kernel Estimator of Invariant Density in Bifurcating Markov Chains Models

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  • S. Valère Bitseki Penda

    (Université de Bourgogne Franche- Comté)

  • Jean-François Delmas

    (École des Ponts)

Abstract

Bifurcating Markov chains are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. Motivated by the functional estimation of the density of the invariant probability measure which appears as the asymptotic distribution of the trait, we prove the consistency and the Gaussian fluctuations for a kernel estimator of this density based on late generations. In this setting, it is interesting to note that the distinction of the three regimes on the ergodic rate identified in a previous work (for fluctuations of average over large generations) disappears. This result is a first step to go beyond the threshold condition on the ergodic rate given in previous statistical papers on functional estimation.

Suggested Citation

  • S. Valère Bitseki Penda & Jean-François Delmas, 2023. "Central Limit Theorem for Kernel Estimator of Invariant Density in Bifurcating Markov Chains Models," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1591-1625, September.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:3:d:10.1007_s10959-022-01205-w
    DOI: 10.1007/s10959-022-01205-w
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    References listed on IDEAS

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    1. S. Valère Bitseki Penda & Angelina Roche, 2020. "Local bandwidth selection for kernel density estimation in a bifurcating Markov chain model," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 32(3), pages 535-562, July.
    2. George Roussas, 1969. "Nonparametric estimation in Markov processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 21(1), pages 73-87, December.
    3. Hoffmann, Marc & Marguet, Aline, 2019. "Statistical estimation in a randomly structured branching population," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5236-5277.
    4. Margarete Knape & Ralph Neininger, 2008. "Approximating Perpetuities," Methodology and Computing in Applied Probability, Springer, vol. 10(4), pages 507-529, December.
    5. 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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