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A law of large numbers result for a bifurcating process with an infinite moving average representation

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  • Terpstra, Jeff T.
  • Elbayoumi, Tamer

Abstract

This paper derives a law of large numbers theorem for bifurcating processes defined on a perfect binary tree. This theorem can be viewed as a generalization of some results that have already appeared in the literature. For instance, all that is required of the bifurcating process is an infinite moving average representation with geometrically decaying coefficients and a finite moment assumption. In addition, the summands are assumed to belong to a flexible class of functions that satisfy a generalized Lipschitz type condition. These two criteria allow for an expansive range of applicability. Two examples are given as corollaries to the theorem.

Suggested Citation

  • Terpstra, Jeff T. & Elbayoumi, Tamer, 2012. "A law of large numbers result for a bifurcating process with an infinite moving average representation," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 123-129.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:1:p:123-129
    DOI: 10.1016/j.spl.2011.09.012
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    References listed on IDEAS

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    1. Huggins, Richard, 1996. "On the identifiability of measurement error in the bifurcating autoregressive model," Statistics & Probability Letters, Elsevier, vol. 27(1), pages 17-23, March.
    2. J. Zhou & I. V. Basawa, 2005. "Maximum Likelihood Estimation for a First‐Order Bifurcating Autoregressive Process with Exponential Errors," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(6), pages 825-842, November.
    3. Zhou, J. & Basawa, I.V., 2005. "Least-squares estimation for bifurcating autoregressive processes," Statistics & Probability Letters, Elsevier, vol. 74(1), pages 77-88, August.
    4. Hwang, S.Y. & Basawa, I.V., 2009. "Branching Markov processes and related asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1155-1167, July.
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