Large deviations for power-law thinned Lévy processes

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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.

Original languageEnglish
Pages (from-to)1353-1384
Number of pages32
JournalStochastic Processes and their Applications
Volume126
Issue number5
DOIs
Publication statusPublished - 1 May 2016

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Random processes
Large Deviations
Power Law
Riemann sum
Rare Events
Tilt
Variational Problem
Stochastic Processes
Random Graphs
Limiting
Choose
Sufficient
First-order
Term

Keywords

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

Cite this

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title = "Large deviations for power-law thinned L{\'e}vy processes",
abstract = "This paper deals with the large deviations behavior of a stochastic process called a thinned L{\'e}vy 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 L{\'e}vy process. General principles prescribe that the tilt should follow from a variational problem, but in the case of the thinned L{\'e}vy 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.",
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Large deviations for power-law thinned Lévy processes. / Aidekon, E.; van der Hofstad, R.W.; Kliem, S.M.; van Leeuwaarden, J.S.H.

In: Stochastic Processes and their Applications, Vol. 126, No. 5, 01.05.2016, p. 1353-1384.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Large deviations for power-law thinned Lévy processes

AU - Aidekon, E.

AU - van der Hofstad, R.W.

AU - Kliem, S.M.

AU - van Leeuwaarden, J.S.H.

PY - 2016/5/1

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N2 - 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.

AB - 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.

KW - Critical random graphs

KW - Exponential tilting

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KW - Thinned Lévy processes

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