Central limit theorem for bifurcating Markov chains: the mother-daughters triangles case

Abstract

The main objective of this article is to establish a central limit theorem for additive three-variable functionals of bifurcating Markov chains. We thus extend the central limit theorem under point-wise ergodic conditions studied in S.V. Bitseki Penda and J.-F. Delmas [Central limit theorem for bifurcating Markov chains under pointwise ergodic conditions, Ann. Appl. Probab. 32(5) (2022), pp. 3817–3849] and to a lesser extent, the results of S.V. Bitseki Penda and J.-F. Delmas [Central limit theorem for bifurcating Markov chains under $L^2$-ergodic conditions, Adv. Appl. Probab. 54(4) (2022), pp. 999–1031] on central limit theorem under $L^2$ ergodic conditions. Our results also extend and complement those of J. Guyon [Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging, Ann. Appl. Probab. 17(5–6) (2007), pp. 1538–1569] and J.-F. Delmas and L. Marsalle [Detection of cellular aging in a Galton-Watson process, Stochastic Process. Appl. 120(12) (2010), pp. 2495–2519]. In particular, when the ergodic rate of convergence is greater than $1/\sqrt{2}$, we have, for certain class of functions, that the asymptotic variance is non-zero at a speed faster than the usual central limit theorem studied by Guyon and Delmas-Marsalle.

Keywords:

• Bifurcating Markov chains
• binary trees
• central limit theorem

Mathematics Subject Classification (2010):

• 60F05
• 60J80

Acknowledgments

The author thanks two anonymous referees for helpful suggestions that led to an improved manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).