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Ergodic theorem for branching markov chains indexed by trees with arbitrary shape

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  09 June 2025

Julien Weibel Open the ORCID record for Julien Weibel [Opens in a new window]
Julien Weibel*
Affiliation:
Institut Denis Poisson and CERMICS
*
*Postal address: Institut Denis Poisson, Université d’Orléans, Université de Tours, CNRS, France and CERMICS, École des Ponts, France. Email: julien.weibel@normalesup.org
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Abstract

We prove an ergodic theorem for Markov chains indexed by the Ulam–Harris–Neveu tree over large subsets with arbitrary shape under two assumptions: (i) with high probability, two vertices in the large subset are far from each other, and (ii) with high probability, those two vertices have their common ancestor close to the root. The assumption on the common ancestor can be replaced by some regularity assumption on the Markov transition kernel. We verify that these assumptions are satisfied for some usual trees. Finally, with Markov chain Monte Carlo considerations in mind, we prove that when the underlying Markov chain is stationary and reversible, the Markov chain, that is the line graph, yields minimal variance for the empirical average estimator among trees with a given number of nodes. In doing so, we prove that the Hosoya–Wiener polynomial is minimized over $[{-}1,1]$ by the line graph among trees of a given size.

Keywords

Tree-indexed Markov chainergodic theoremBienaymé–Galton–Watson treesminimal variance

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F25: $L^p$-limit theorems
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 16
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Abraham, R. and Delmas, J.-F. (2020). An introduction to Galton–Watson trees and their local limits. Preprint, arXiv:1506.05571.Google Scholar
Athreya, K. B. (2012). Coalescence in critical and subcritical Galton–Watson branching. J. Appl. Prob. 49, 627–638.10.1239/jap/1346955322CrossRefGoogle Scholar
Athreya, K. B. (2012). Coalescence in the recent past in rapidly growing populations. Stoch. Process. Appl. 122, 3757–3766.10.1016/j.spa.2012.06.015CrossRefGoogle Scholar
Athreya, K. B. and Kang, H.-J. (1998). Some limit theorems for positive recurrent branching Markov chains: I. Adv. Appl. Prob. 30, 693–710.10.1239/aap/1035228124CrossRefGoogle Scholar
Athreya, K. B. and Kang, H.-J. (1998). Some limit theorems for positive recurrent branching Markov chains: II. Adv. Appl. Prob. 30, 711–722.10.1239/aap/1035228125CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes, Springer, Berlin.10.1007/978-3-642-65371-1CrossRefGoogle Scholar
Bansaye, V. (2019). Ancestral lineages and limit theorems for branching Markov chains in varying environment. J. Theoret. Prob. 32, 249–281.10.1007/s10959-018-0825-1CrossRefGoogle Scholar
Casablanca, R. M. and Dankelmann, P. (2019). Distance and eccentric sequences to bound the Wiener index, Hosoya polynomial and the average eccentricity in the strong products of graphs. Discrete Appl. Math. 263, 105–117.10.1016/j.dam.2018.07.009CrossRefGoogle Scholar
Delmas, J.-F. and Marsalle, L. (2010). Detection of cellular aging in a Galton–Watson process. Stoch. Process. Appl. 120, 2495–2519.10.1016/j.spa.2010.07.002CrossRefGoogle Scholar
Guyon, J. (2007). Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Prob. 17, 1538–1569.10.1214/105051607000000195CrossRefGoogle Scholar
Rudin, W. (1996). Functional Analysis, 2nd edn. McGraw-Hill, Boston, MA.Google Scholar
Tian, D. and Choi, K. P. (2013). Sharp bounds and normalization of Wiener-type indices. PLoS ONE 8, e78448.10.1371/journal.pone.0078448CrossRefGoogle ScholarPubMed
Wagner, S., Wang, H. and Zhang, X.-D. (2013). Distance-based graph invariants of trees and the Harary index, Filomat 1, 41–50.10.2298/FIL1301041WCrossRefGoogle Scholar