Electronic Communications in Probability

Some limit results for Markov chains indexed by trees

Peter Czuppon and Peter Pfaffelhuber

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Abstract

We consider a sequence of Markov chains $(\mathcal X^n)_{n=1,2,...}$ with $\mathcal X^n = (X^n_\sigma)_{\sigma\in\mathcal T}$, indexed by the full binary tree $\mathcal T = \mathcal T_0 \cup \mathcal T_1 \cup ...$, where $\mathcal T_k$ is the $k$th generation of $\mathcal T$. In addition, let $(\Sigma_k)_{k=0,1,2,...}$ be a random walk on $\mathcal T$ with $\Sigma_k \in \mathcal T_k$ and $\widetilde{\mathcal R}^n = (\widetilde R_t^n)_{t\geq 0}$ with $\widetilde R_t^n := X_{\Sigma_{[tn]}}$, arising by observing the Markov chain $\mathcal X^n$ along the random walk. We present a law of large numbers concerning the empirical measure process $\widetilde{\mathcal Z}^n = (\widetilde Z_t^n)_{t\geq 0}$ where $\widetilde{Z}_t^n = \sum_{\sigma\in\mathcal T_{[tn]}} \delta_{X_\sigma^n}$ as $n\to\infty$. Precisely, we show that if $\widetilde{\mathcal R}^n \Rightarrow{n\to\infty} \mathcal R$ for some Feller process $\mathcal R = (R_t)_{t\geq 0}$ with deterministic initial condition, then $\widetilde{\mathcal Z}^n \Rightarrow{n\to\infty} \mathcal Z$ with $Z_t = \delta_{\mathcal L(R_t)}$.

Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 77, 11 pp.

Dates
Accepted: 11 November 2014
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465316779

Digital Object Identifier
doi:10.1214/ECP.v19-3601

Mathematical Reviews number (MathSciNet)
MR3283608

Zentralblatt MATH identifier
1334.60036

Subjects
Primary: 60F15: Strong theorems
Secondary: 60F05: Central limit and other weak theorems

Keywords
Tree-indexed Markov chain weak convergence tightness random measure empirical measure

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Czuppon, Peter; Pfaffelhuber, Peter. Some limit results for Markov chains indexed by trees. Electron. Commun. Probab. 19 (2014), paper no. 77, 11 pp. doi:10.1214/ECP.v19-3601. https://projecteuclid.org/euclid.ecp/1465316779


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References

  • S.V. Avery. Microbial cell individuality and the underlying sources of heterogeneity. phNat. Rev. Microbiol., 4:577–587, 2006.
  • Benjamini, Itai; Peres, Yuval. Markov chains indexed by trees. Ann. Probab. 22 (1994), no. 1, 219–243.
  • Bercu, Bernard; de Saporta, Benoîte; Gégout-Petit, Anne. Asymptotic analysis for bifurcating autoregressive processes via a martingale approach. Electron. J. Probab. 14 (2009), no. 87, 2492–2526.
  • Dawson, Donald A. Measure-valued Markov processes. École d'Été de Probabilités de Saint-Flour XXI—1991, 1–260, Lecture Notes in Math., 1541, Springer, Berlin, 1993.
  • Delmas, Jean-François; Marsalle, Laurence. Detection of cellular aging in a Galton-Watson process. Stochastic Process. Appl. 120 (2010), no. 12, 2495–2519.
  • de Saporta, Benoîte; Gégout-Petit, Anne; Marsalle, Laurence. Parameters estimation for asymmetric bifurcating autoregressive processes with missing data. Electron. J. Stat. 5 (2011), 1313–1353.
  • Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8
  • M.B. Elowitz, A.J. Levine, E.D. Siggia, and P. Swain. Stochastic gene expression in a single cell. phScience Signalling, 297:1183–1186, 2002.
  • Guyon, Julien. Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Probab. 17 (2007), no. 5-6, 1538–1569.
  • Jakubowski, Adam. On the Skorokhod topology. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 3, 263–285.
  • Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2
  • Liu, Wen; Wang, Liying. The Markov approximation of the random fields on Cayley trees and a class of small deviation theorems. Statist. Probab. Lett. 63 (2003), no. 2, 113–121.
  • Liu, Wen; Yang, Weiguo. Some strong limit theorems for Markov chain fields on trees. Probab. Engrg. Inform. Sci. 18 (2004), no. 3, 411–422.
  • H.H. McAdams and A. Arkin. It’s a noisy business! genetic regulation at the nanomolar scale. phTrends Genet., 15:65–69, 1999.
  • L. Pelkmans. Using cell-to-cell variability – a new era in molecular biology. phScience, 336:425–426, 2012.
  • J.L. Spudich and D.E. Koshland. Non-genetic individuality: chance in the single cell. phNature, 262:467–471, 1976.
  • B. Snijder and L. Pelkmans. Origins of regulated cell-to-cell variability. phNat. Rev. Mol. Cell. Biol., 12:119–125, 2011.
  • Takacs, Christiane. On the fundamental matrix of finite state Markov chains, its eigensystem and its relation to hitting times. Math. Pannon. 17 (2006), no. 2, 183–193.
  • Yang, Weiguo. Some limit properties for Markov chains indexed by a homogeneous tree. Statist. Probab. Lett. 65 (2003), no. 3, 241–250.
  • Yang, Weiguo; Liu, Wen. Strong law of large numbers and Shannon-McMillan theorem for Markov chain fields on trees. IEEE Trans. Inform. Theory 48 (2002), no. 1, 313–318.