Large deviations for power-law thinned Levy processes

作者：Aidekon Elie; van der Hofstad Remco; Kliem Sandra; van Leeuwaarden Johan S H^*

来源：Stochastic Processes and Their Applications, 2016, 126(5): 1353-1384.

DOI：10.1016/j.spa.2015.11.006

• Thinned Levy processes
• Large deviations
• Exponential tilting
• Critical random graphs

This paper deals with the large deviations behavior of a stochastic process called a thinned Levy 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 Levy process. General principles prescribe that the tilt should follow from a variational problem, but in the case of the thinned Levy 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.

• 出版日期2016-5