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On the functional and local limit theorems for Markov modulated compound Poisson processes

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  • Pang, Guodong
  • Zheng, Yi

Abstract

We study a class of Markov-modulated compound Poisson processes whose arrival rates and the compound random variables are both modulated by a stationary finite-state Markov process. The compound random variables are i.i.d. in each state of the Markov process, while having a distribution depending on the state of the Markov process. We prove a functional central limit theorem and local limit theorems under appropriate scalings of the arrival process, compound random variables and underlying Markov process.

Suggested Citation

  • Pang, Guodong & Zheng, Yi, 2017. "On the functional and local limit theorems for Markov modulated compound Poisson processes," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 131-140.
    • Handle: RePEc:eee:stapro:v:129:y:2017:i:c:p:131-140
    DOI: 10.1016/j.spl.2017.05.009
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    References listed on IDEAS

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    1. Giuliano-Antonini, Rita & Szewczak, Zbigniew S., 2013. "An almost sure local limit theorem for Markov chains," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 573-579.
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    4. Hongyuan Lu & Guodong Pang & Michel Mandjes, 2016. "A functional central limit theorem for Markov additive arrival processes and its applications to queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 84(3), pages 381-406, December.
    5. D. Anderson & J. Blom & M. Mandjes & H. Thorsdottir & K. Turck, 2016. "A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 153-168, March.
    6. Søren Asmussen & Colm O'cinneide, 2002. "On the Tail of the Waiting Time in a Markov-Modulated M/G/1 Queue," Operations Research, INFORMS, vol. 50(3), pages 559-565, June.
    7. Ward Whitt, 2016. "Heavy-traffic fluid limits for periodic infinite-server queues," Queueing Systems: Theory and Applications, Springer, vol. 84(1), pages 111-143, October.
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    Cited by:

    1. G. A. Delsing & M. R. H. Mandjes & P. J. C. Spreij & E. M. M. Winands, 2020. "Asymptotics and Approximations of Ruin Probabilities for Multivariate Risk Processes in a Markovian Environment," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 927-948, September.
    2. Sen, Ankita & Selvaraju, N., 2023. "Diffusion approximation of an infinite-server queue under Markovian environment with rapid switching," Statistics & Probability Letters, Elsevier, vol. 195(C).

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