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Large deviations for a randomly indexed branching process with immigration

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  • Gao, Zhenlong

Abstract

Consider a supercritical continuous time branching process called randomly indexed branching processes with immigration. Large deviation results are established for the logarithms of such processes. Our results show that when the offspring distribution belongs to the Schröder case, the immigration distribution affects the rate function of the large deviation, while when the offspring distribution belongs to the Böttcher case, the immigration distribution has no effect on the rate function.

Suggested Citation

  • Gao, Zhenlong, 2026. "Large deviations for a randomly indexed branching process with immigration," Statistics & Probability Letters, Elsevier, vol. 227(C).
    • Handle: RePEc:eee:stapro:v:227:y:2026:i:c:s0167715225001919
    DOI: 10.1016/j.spl.2025.110546
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    References listed on IDEAS

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    1. Gao, Zhenlong & Zhang, Yanhua, 2015. "Large and moderate deviations for a class of renewal random indexed branching process," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 1-5.
    2. Huang, Chunmao & Liu, Quansheng, 2012. "Moments, moderate and large deviations for a branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 522-545.
    3. Gao, Zhenlong & Wang, Weigang, 2016. "Large and moderate deviations for a renewal randomly indexed branching process," Statistics & Probability Letters, Elsevier, vol. 116(C), pages 139-145.
    4. Gao, Zhenlong & Wang, Weigang, 2015. "Large deviations for a Poisson random indexed branching process," Statistics & Probability Letters, Elsevier, vol. 105(C), pages 143-148.
    5. Mitov, Georgi K. & Mitov, Kosto V. & Yanev, Nikolay M., 2009. "Critical randomly indexed branching processes," Statistics & Probability Letters, Elsevier, vol. 79(13), pages 1512-1521, July.
    6. Zhengyan Lin & Zhidong Bai, 2010. "Probability Inequalities of Random Variables," Springer Books, in: Probability Inequalities, chapter 0, pages 37-50, Springer.
    7. Georgi K. Mitov & Svetlozar T. Rachev & Young Shin Kim & Frank J. Fabozzi, 2009. "Barrier Option Pricing By Branching Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(07), pages 1055-1073.
    8. Zhengyan Lin & Zhidong Bai, 2010. "Elementary Inequalities of Probabilities of Events," Springer Books, in: Probability Inequalities, chapter 0, pages 1-8, Springer.
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