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Functional limit theorems for a new class of non-stationary shot noise processes

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  • Pang, Guodong
  • Zhou, Yuhang

Abstract

We study a class of non-stationary shot noise processes which have a general arrival process of noises with non-stationary arrival rate and a general shot shape function. Given the arrival times, the shot noises are conditionally independent and each shot noise has a general (multivariate) cumulative distribution function (c.d.f.) depending on its arrival time. We prove a functional weak law of large numbers and a functional central limit theorem for this new class of non-stationary shot noise processes in an asymptotic regime with a high intensity of shot noises, under some mild regularity conditions on the shot shape function and the conditional (multivariate) c.d.f. We discuss the applications to a simple multiplicative model (which includes a class of non-stationary compound processes and applies to insurance risk theory and physics) and the queueing and work-input processes in an associated non-stationary infinite-server queueing system. To prove the weak convergence, we show new maximal inequalities and a new criterion of existence of a stochastic process in the space D given its consistent finite dimensional distributions, which involve a finite set function with the superadditive property.

Suggested Citation

  • Pang, Guodong & Zhou, Yuhang, 2018. "Functional limit theorems for a new class of non-stationary shot noise processes," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 505-544.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:2:p:505-544
    DOI: 10.1016/j.spa.2017.05.008
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    References listed on IDEAS

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    1. 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.
    2. Iksanov, Alexander, 2013. "Functional limit theorems for renewal shot noise processes with increasing response functions," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1987-2010.
    3. Iglehart, Donald L., 1973. "Weak convergence of compound stochastic process, I," Stochastic Processes and their Applications, Elsevier, vol. 1(1), pages 11-31, January.
    4. Pang, Guodong & Zhou, Yuhang, 2017. "Two-parameter process limits for an infinite-server queue with arrival dependent service times," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1375-1416.
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    Cited by:

    1. Li, Bo & Pang, Guodong, 2022. "Functional limit theorems for nonstationary marked Hawkes processes in the high intensity regime," Stochastic Processes and their Applications, Elsevier, vol. 143(C), pages 285-339.

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