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Asymptotic deviation bounds for cumulative processes

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  • Cattiaux, Patrick
  • Colombani, Laetitia
  • Costa, Manon

Abstract

The aim of this paper is to get asymptotic deviation bounds via a Large Deviation Principle (LDP) for cumulative processes also known as compound renewal processes or renewal-reward processes. These processes cumulate independent random variables occurring in time interval given by a renewal process. Our result extends the one obtained in Lefevere et al. (2011) in the sense that we impose no specific dependency between the cumulated random variables and the renewal process and the proof uses Mariani and Zambotti (2014). In the companion paper Cattiaux et al. (2022) we apply this principle to Hawkes processes with inhibition. Under some assumptions Hawkes processes are indeed cumulative processes, but they do not enter the framework of Lefevere et al. (2011).

Suggested Citation

  • Cattiaux, Patrick & Colombani, Laetitia & Costa, Manon, 2023. "Asymptotic deviation bounds for cumulative processes," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 85-105.
    • Handle: RePEc:eee:spapps:v:163:y:2023:i:c:p:85-105
    DOI: 10.1016/j.spa.2023.05.010
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    References listed on IDEAS

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    1. Zamparo, Marco, 2021. "Large deviations in discrete-time renewal theory," Stochastic Processes and their Applications, Elsevier, vol. 139(C), pages 80-109.
    2. Tiefeng, Jiang, 1994. "Large deviations for renewal processes," Stochastic Processes and their Applications, Elsevier, vol. 50(1), pages 57-71, March.
    3. Lefevere, Raphaël & Mariani, Mauro & Zambotti, Lorenzo, 2011. "Large deviations for renewal processes," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2243-2271, October.
    4. Cattiaux, Patrick & Colombani, Laetitia & Costa, Manon, 2022. "Limit theorems for Hawkes processes including inhibition," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 404-426.
    5. Zamparo, Marco, 2023. "Large deviation principles for renewal–reward processes," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 226-245.
    6. Glynn, Peter W. & Whitt, Ward, 1993. "Limit theorems for cumulative processes," Stochastic Processes and their Applications, Elsevier, vol. 47(2), pages 299-314, September.
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