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A central limit theorem for martingales and an application to branching processes

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  • Scott, D. J.

Abstract

A functional central limit theorem is obtained for martingales which are not uniformly asymptotically negligible but grow at a geometric rate. The function space is not the usual C[0,1] or D[0,1] but RN, the space of all real sequences and the metric used leads to a non-separable metric space. The main theorem is applied to a martingale obtained from a supercritical Galton-Watson branching process and as simple corollaries the already known central limit theorems for the Harris and Lotka-Nagaev estimators of the mean of the offspring distribution, are obtained.

Suggested Citation

  • Scott, D. J., 1978. "A central limit theorem for martingales and an application to branching processes," Stochastic Processes and their Applications, Elsevier, vol. 6(3), pages 241-252, February.
    • Handle: RePEc:eee:spapps:v:6:y:1978:i:3:p:241-252
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    Cited by:

    1. James Kuelbs & Anand N. Vidyashankar, 2011. "Weak Convergence Results for Multiple Generations of a Branching Process," Journal of Theoretical Probability, Springer, vol. 24(2), pages 376-396, June.
    2. Rahimov, I., 2017. "Asymptotic inference for non-supercritical partially observed branching processes," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 26-32.
    3. Arpita Inamdar & Mohan Kale, 2016. "Joint Estimation of Offspring Mean and Offspring Variance of Controlled Branching Process," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 78(2), pages 248-268, August.
    4. Horst, Ulrich & Xu, Wei, 2021. "Functional limit theorems for marked Hawkes point measures," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 94-131.

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