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Large Time Asymptotic of Heavy Tailed Renewal Processes

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Abstract

We study the large-time asymptotic of renewal-reward processes with a heavy-tailed waiting time distribution. It is known that the heavy tail of the distribution produces an extremely slow dynamics, resulting in a singular large deviation function. When the singularity takes place, the bottom of the large deviation function is flattened, manifesting anomalous fluctuations of the renewal-reward processes. In this article, we aim to study how these singularities emerge as the time increases. Using a classical result on the sum of random variables with regularly varying tail, we develop an expansion approach to prove an upper bound of the finite-time moment generating function for the Pareto waiting time distribution (power law) with an integer exponent. We perform numerical simulations using Pareto (with a real value exponent), inverse Rayleigh and log-normal waiting time distributions, and demonstrate similar results are anticipated in these waiting time distributions.

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Acknowledgements

H.H. is in the Cofund MathInParis PhD project that has received funding from the European Union’s Horizon 2020 research and agreement No. 754362.  R.L. is supported by the ANR-15-CE40-0020-01 grant LSD. The authors thank Robert Jack for fruitful discussions about finite-time CGFs and rate functions. H.H. also acknowledges the support from Robert Jack during his stay in the University of Cambridge for his mobility project.

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Authors and Affiliations

  1. Université de Paris, Laboratoire de Probabilités, Statistiques et Modélisation, UMR 8001, F-75205, Paris, France

    Hiroshi Horii & Raphaël Lefevere

  2. Sorbonne Université, INSERM, CNRS, Institut de la Vision, 75012, Paris, France

    Takahiro Nemoto

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  1. Hiroshi Horii
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  2. Raphaël Lefevere
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  3. Takahiro Nemoto
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Correspondence to Raphaël Lefevere.

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Communicated by Cristina Toninelli.

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Horii, H., Lefevere, R. & Nemoto, T. Large Time Asymptotic of Heavy Tailed Renewal Processes. J Stat Phys 186, 11 (2022). https://doi.org/10.1007/s10955-021-02856-5

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