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Asymptotics for irregularly observed long memory processes

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  • Ould Haye, Mohamedou
  • Philippe, Anne

Abstract

We study the effect of observing a long-memory stationary process at irregular time points via a renewal process. We establish a sharp difference in the asymptotic behaviour of the self-normalized sample mean of the observed process depending on the renewal process. In particular, we show that if the renewal process has a moderate heavy-tail distribution, then the limit is a so-called Normal Variance Mixture (NVM) and we characterize the randomized variance part of the limiting NVM as an integral function of a Lévy stable motion. Otherwise, the normalized sample mean will be asymptotically normal.

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  • Ould Haye, Mohamedou & Philippe, Anne, 2025. "Asymptotics for irregularly observed long memory processes," Stochastic Processes and their Applications, Elsevier, vol. 185(C).
    • Handle: RePEc:eee:spapps:v:185:y:2025:i:c:s0304414925000729
    DOI: 10.1016/j.spa.2025.104631
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    References listed on IDEAS

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    1. Henghsiu Tsai & K. S. Chan, 2005. "Quasi‐Maximum Likelihood Estimation for a Class of Continuous‐time Long‐memory Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(5), pages 691-713, September.
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    3. F. Comte, 1996. "Simulation And Estimation Of Long Memory Continuous Time Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 17(1), pages 19-36, January.
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    7. Mohamedou Ould Haye & Anne Philippe & Caroline Robet, 2024. "Inference for continuous-time long memory randomly sampled processes," Statistical Papers, Springer, vol. 65(5), pages 3111-3134, July.
    8. Cremers, Heinz & Kadelka, Dieter, 1986. "On weak convergence of integral functionals of stochastic processes with applications to processes taking paths in LEP," Stochastic Processes and their Applications, Elsevier, vol. 21(2), pages 305-317, February.
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