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Extremes

, Volume 19, Issue 3, pp 371–403 | Cite as

Sup-norm convergence rates for Lévy density estimation

  • Valentin Konakov
  • Vladimir PanovEmail author
Article

Abstract

In this paper, we consider projection estimates for Lévy densities in high-frequency setup. We give a unified treatment for different sets of basis functions and focus on the asymptotic properties of the maximal deviation distribution for these estimates. Our results are based on the idea to reformulate the problems in terms of Gaussian processes of some special type and to further analyze these Gaussian processes. In particular, we construct a sequence of excursion sets, which guarantees the convergence of the deviation distribution to the Gumbel distribution. We show that the exact rates of convergence presented in previous articles on this topic are logarithmic and construct the sequence of accompanying laws, which approximate the deviation distribution with polynomial rate.

Keywords

Lévy density Maximal deviation Nonparametric inference Projection estimates 

AMS 2000 Subject Classifications

60G51 62M99 62G05 

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References

  1. Belomestny, D.: Statistical inference for time-changed Lévy processes via composite characteristic function estimation. Ann. Stat. 39(4), 2205–2242 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bickel, P., Rosenblatt, M.: On some global measures of the deviations of density function estimates. Ann. Stat. 1(6), 1071–1095 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chen, S.X., Delaigle, A., Hall, P.: Nonparametric estimation for a class of Lévy processes. J. Econ. 157, 257–271 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Comte, F., Genon-Catalot, V.: Nonparametric estimation for pure jump Lévy processes based on high frequency data. Stoch. Process. Appl. 119, 4088–4123 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Comte, F., Genon-Catalot, V.: Nonparametric estimation for pure jump irregularly sampled or noisy Lévy processes. Statistica Neerlandica 64, 290–313 (2010a)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Comte, F., Genon-Catalot, V.: Nonparametric adaptive estimation for pure jump Lévy processes. Ann. l’Inst. Henri Poincaré - Prob. Stat. 46(3), 595–617 (2010b)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Comte, F., Genon-Catalot, V.: Estimation for Lévy processes from high frequency data within a long time interval. Ann. Stat. 39, 803–837 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Figueroa-López, J.E.: Nonparametric Estimation of Lévy Processes with a View Towards Mathematical Finance. PhD thesis, Georgia Institute of Technology (2004)Google Scholar
  9. Figueroa-López, J.E.: Sieve-based confidence intervals and bands for Lévy densities. Bernoulli 17(2), 643–670 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Fisher, M., Nappo, G.: On the moments of the modulus of continuity of Ito Processes. Stochast. Process. Appl. 28(1), 103–122 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Gradshtein, I., Ryzhik I.: Table of integrals, Series and Products. Academic Press (1996)Google Scholar
  12. Gugushvili, S.: Nonparametric inference for discretely sampled Lévy processes. Ann. l’Inst. Henri Poincaré - Probab. Stat. 48(1), 282–307 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hall, P.: On convergences rates of suprema. Probab. Theory Relat. Fields 89 (447-455) (1991)Google Scholar
  14. Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent rv’s and the sample DF. Z. Wahrscheinlichkeitstheorie Verw Geb. 32, 111–131 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Konakov, V., Piterbarg, V.: On the convergence rates of maximal deviation distribution for kernel regression estimates. J. Multivar. Anal. 15, 279–294 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Konakov, V., Panov, V.: Convergence rates of maximal deviation distribution for projection estimates of Lévy densities (2016). arXiv:1411.4750v3
  17. Kuo, H.-H.: Introduction to stochastic integration. Springer (2006)Google Scholar
  18. Michna, Z.: Remarks on Pickands theorem (2009). arXiv:0904.3832v1
  19. Neumann, M., Reiss, M.: Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15(1), 223–248 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Nickl, R., Reiss, M.: A Donsker theorem for Lévy measures. J. Funct. Anal. 253, 3306–3332 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Piterbarg, V.I.: Asymptotic Methods in the Theory of Gaussian Processes and Fields. AMS, Providence (1996)Google Scholar
  22. Piterbarg, V.I.: Twenty Lectures about Gaussian Processes. Atlantic Financial Press, London (2015)zbMATHGoogle Scholar
  23. Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Springer-Verlag (1999)Google Scholar
  24. Sato, K.: Lévy processes and infinitely divisible distributions. Cambridge University Press (1999)Google Scholar
  25. Suetin, P.: Classical orthogonal polynomials (in Russian). Fizmatlit (2005)Google Scholar
  26. Van Es, B., Gugushvili, S., Spreij, P.: A kernel type nonparametric density estimator for decompounding. Bernoulli 13, 672–694 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Wörner, J.: Variational sums and power variations: a unifying approach to model selection and estimation in semimartingale models. Stat. Decis. 21, 47–68 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia

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