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Large deviations for power-law thinned Lévy processes

Author

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  • Aïdékon, Elie
  • van der Hofstad, Remco
  • Kliem, Sandra
  • van Leeuwaarden, Johan S.H.

Abstract

This paper deals with the large deviations behavior of a stochastic process called a thinned Lévy process. This process appeared recently as a stochastic-process limit in the context of critical inhomogeneous random graphs (Bhamidi et al. (2012)). The process has a strong negative drift, while we are interested in the rare event of the process being positive at large times. To characterize this rare event, we identify a tilted measure. This presents some challenges inherent to the power-law nature of the thinned Lévy process. General principles prescribe that the tilt should follow from a variational problem, but in the case of the thinned Lévy process this involves a Riemann sum that is hard to control. We choose to approximate the Riemann sum by its limiting integral, derive the first-order correction term, and prove that the tilt that follows from the corresponding approximate variational problem is sufficient to establish the large deviations results.

Suggested Citation

  • Aïdékon, Elie & van der Hofstad, Remco & Kliem, Sandra & van Leeuwaarden, Johan S.H., 2016. "Large deviations for power-law thinned Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 126(5), pages 1353-1384.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:5:p:1353-1384
    DOI: 10.1016/j.spa.2015.11.006
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