Sample path large deviations for levy processes and random walks with regularly varying increments

Chang Han Rhee*, Jose Blanchet, Bert Zwart

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

Let X be a Levy process with regularly varying Levy measure ν. We obtain sample-path large deviations for scaled processes. Xn(t) X(nt)/n and obtain a similar result for random walks with regularly varying increments. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the increment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connections with the classical large deviations framework. In that setting, we show that a weak large deviation principle (with logarithmic speed) holds, but a full large deviation principle does not hold.

Original languageEnglish (US)
Pages (from-to)3551-3605
Number of pages55
JournalAnnals of Probability
Volume47
Issue number6
DOIs
StatePublished - 2019

Keywords

  • Lévy processes
  • M-convergence
  • Regular variation
  • Sample path large deviations

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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