Large deviations for power-law thinned Lévy processes

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

Stochastic Processes and their Applications, 2016, vol. 126, issue 5, 1353-1384

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.

Keywords: Thinned Lévy processes; Large deviations; Exponential tilting; Critical random graphs (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:126:y:2016:i:5:p:1353-1384

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DOI: 10.1016/j.spa.2015.11.006

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