Asymptotic Analysis for Bifurcating AutoRegressive Processes via a Martingale Approach

Bernard Bercu (Université de Bordeaux)
Benoîte de Saporta (Université de Bordeaux)
Anne Gégout-Petit (Université de Bordeaux)


We study the asymptotic behavior of the least squares estimators of the unknown parameters of general pth-order bifurcating autoregressive processes. Under very weak assumptions on the driven noise of the process, namely conditional pair-wise independence and suitable moment conditions, we establish the almost sure convergence of our estimators together with the quadratic strong law and the central limit theorem. All our analysis relies on non-standard asymptotic results for martingales.

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Pages: 2492-2526

Publication Date: November 11, 2009

DOI: 10.1214/EJP.v14-717


  1. Basawa, I. V.; Zhou, J. Non-Gaussian bifurcating models and quasi-likelihood estimation.Stochastic methods and their applications. J. Appl. Probab. 41A (2004), 55--64. MR2057565
  2. Cowan, R., and Staudte, R.~G. The bifurcating autoregressive model in cell lineage studies. Biometrics 42 (1986), 769--783.
  3. Delmas, J.-F., and Marsalle, L. Detection of cellular aging in a Galton-Watson process. arXiv, 0807.0749 (2008).
  4. Duflo, Marie. Random iterative models.Translated from the 1990 French original by Stephen S. Wilson and revised by the author.Applications of Mathematics (New York), 34. Springer-Verlag, Berlin, 1997. xviii+385 pp. ISBN: 3-540-57100-0 MR1485774 (98m:62239)
  5. 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. MR2358633 (2009e:60051)
  6. Guyon, Julien; Bize, Ariane; Paul, Grégory; Stewart, Eric; Delmas, Jean-Francois; Taddéi, Francois. Statistical study of cellular aging. CEMRACS 2004---mathematics and applications to biology and medicine, 100--114 (electronic), ESAIM Proc., 14, EDP Sci., Les Ulis, 2005. MR2226805
  7. Hall, P.; Heyde, C. C. Martingale limit theory and its application.Probability and Mathematical Statistics.Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. xii+308 pp. ISBN: 0-12-319350-8 MR0624435 (83a:60001)
  8. Hamilton, James D. Time series analysis.Princeton University Press, Princeton, NJ, 1994. xvi+799 pp. ISBN: 0-691-04289-6 MR1278033 (95h:62165)
  9. Huggins, R. M. Robust inference for variance components models for single trees of cell lineage data. Ann. Statist. 24 (1996), no. 3, 1145--1160. MR1401842 (97k:62065)
  10. Huggins, R. M.; Basawa, I. V. Extensions of the bifurcating autoregressive model for cell lineage studies. J. Appl. Probab. 36 (1999), no. 4, 1225--1233. MR1746406
  11. Huggins, R. M.; Basawa, I. V. Inference for the extended bifurcating autoregressive model for cell lineage studies. Aust. N. Z. J. Stat. 42 (2000), no. 4, 423--432. MR1802966 (2002d:62061)
  12. Hwang, S.~Y., Basawa, I.~V., and Yeo, I.~K. Local asymptotic normality for bifurcating autoregressive processes and related asymptotic inference. Statistical Methodology 6 (2009), 61--69.
  13. Wei, C. Z. Adaptive prediction by least squares predictors in stochastic regression models with applications to time series. Ann. Statist. 15 (1987), no. 4, 1667--1682. MR0913581 (89e:62123)
  14. Zhou, J.; Basawa, I. V. Least-squares estimation for bifurcating autoregressive processes. Statist. Probab. Lett. 74 (2005), no. 1, 77--88. MR2189078 (2006m:62073)
  15. Zhou, J.; Basawa, I. V. Maximum likelihood estimation for a first-order bifurcating autoregressive process with exponential errors. J. Time Ser. Anal. 26 (2005), no. 6, 825--842. MR2203513 (2006m:62026)

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