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Working paper

Convergence rates of maximal deviation distribution for projection estimates of L\'evy densities

arxiv.org. math. Cornell University, 2014. No. 1411.4750.
In this paper, we consider projection estimates for L{\'e}vy 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 maximal deviation distribution to the Gumbel distribution. We show that the rates of convergence presented in previous research on this topic are logarithmic and construct the sequences of accompanying laws with power rates.