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 language | English |
|---|---|
| Pages (from-to) | 1353-1384 |
| Number of pages | 32 |
| Journal | Stochastic Processes and their Applications |
| Volume | 126 |
| Issue number | 5 |
| DOIs | |
| Publication status | Published - 1 May 2016 |
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Keywords
- Critical random graphs
- Exponential tilting
- Large deviations
- Thinned Lévy processes
Cite this
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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 journal › Article › Academic › peer-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
Y1 - 2016/5/1
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
KW - Large deviations
KW - Thinned Lévy processes
UR - http://www.scopus.com/inward/record.url?scp=84960802270&partnerID=8YFLogxK
U2 - 10.1016/j.spa.2015.11.006
DO - 10.1016/j.spa.2015.11.006
M3 - Article
AN - SCOPUS:84960802270
VL - 126
SP - 1353
EP - 1384
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
SN - 0304-4149
IS - 5
ER -