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.
Lévy density Maximal deviation Nonparametric inference Projection estimates
AMS 2000 Subject Classifications
60G51 62M99 62G05
This is a preview of subscription content, log in to check access.
Hall, P.: On convergences rates of suprema. Probab. Theory Relat. Fields 89 (447-455) (1991)Google Scholar
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
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
Konakov, V., Panov, V.: Convergence rates of maximal deviation distribution for projection estimates of Lévy densities (2016). arXiv:1411.4750v3
Kuo, H.-H.: Introduction to stochastic integration. Springer (2006)Google Scholar
Michna, Z.: Remarks on Pickands theorem (2009). arXiv:0904.3832v1