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Large and Moderate Deviations for the Total Population Arising from a Sub-critical Galton–Watson Process with Immigration

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Listed:
  • Shihang Yu

    (Qiqihar University)

  • Dehui Wang

    (Jilin University)

  • Xia Chen

    (University of Tennessee)

Abstract

In this paper, we provide the exact forms of large and moderate deviations for the empirical mean of population and the centered total population of a sub-critical branching process with immigration. The rate functions in our large and moderate deviations are explicitly identified. Our theorems also apply to the models of the integer-valued autoregression. In computing the generating function requested by Gärtner-Ellis theorem, our treatment substantially relies on an algorithm specifically designed for the autoregressive structure of our models.

Suggested Citation

  • Shihang Yu & Dehui Wang & Xia Chen, 2018. "Large and Moderate Deviations for the Total Population Arising from a Sub-critical Galton–Watson Process with Immigration," Journal of Theoretical Probability, Springer, vol. 31(1), pages 41-67, March.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:1:d:10.1007_s10959-016-0706-4
    DOI: 10.1007/s10959-016-0706-4
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    References listed on IDEAS

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    1. Christian Weiß, 2008. "Thinning operations for modeling time series of counts—a survey," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 92(3), pages 319-341, August.
    2. A. Alzaid & M. Al‐Osh, 1988. "First‐Order Integer‐Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 42(1), pages 53-61, March.
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