[Paper Review] Convergence rates of maximal deviation distribution for projection estimates of Lévy densities

Valentin Konakov, Vladimir Panov|arXiv (Cornell University)|Nov 18, 2014

Stochastic processes and financial applications20 references3 citations

TL;DR

This paper establishes polynomial convergence rates for the maximal deviation distribution of projection estimators of Lévy densities in a high-frequency setting, using a unified framework based on Gaussian process approximation and excursion set analysis. It proves that the limiting distribution of the normalized maximal deviation converges to the Gumbel distribution with polynomial speed, improving upon prior logarithmic rates.

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 rates of convergence presented in previous articles on this topic are logarithmic and construct the sequences of accompanying laws, which approximate the deviation distribution with polynomial rate.

Motivation & Objective

• To analyze the asymptotic behavior of the maximal deviation distribution for projection estimators of Lévy densities in high-frequency settings.
• To unify the treatment of different basis functions—trigonometric, Legendre, and wavelet bases—under a common analytical framework.
• To improve upon existing logarithmic convergence rates by constructing sequences of accompanying laws that achieve polynomial convergence speed.
• To establish the convergence of the maximal deviation distribution to the Gumbel distribution through the construction of appropriate excursion sets.
• To provide a non-asymptotic analysis of the deviation between the estimator and the true density, incorporating bias and variance components.

Proposed method

• Reformulate the maximal deviation problem in terms of a Gaussian process indexed by the basis functions, enabling the use of extreme value theory.
• Construct a sequence of excursion sets for the Gaussian process that ensures convergence of the deviation distribution to the Gumbel distribution.
• Use the Pickands theorem and Taylor expansions to analyze the asymptotic behavior of the first passage time of the process.
• Apply local homogeneity and global Hölder conditions on the covariance function to control the sample path behavior of the process.
• Derive bias and variance bounds using small-time asymptotics of the Lévy process and moment conditions on the basis functions.
• Establish uniform bounds on the deviation using concentration inequalities and the structure of the basis functions (e.g., Legendre polynomials, wavelets).

Experimental results

Research questions

• RQ1 What is the rate of convergence of the distribution of the maximal deviation between the projection estimator and the true Lévy density?
• RQ2 Can the limiting distribution of the normalized maximal deviation be shown to converge to the Gumbel distribution under general basis functions?
• RQ3 How do the convergence rates compare between existing logarithmic rates and the proposed polynomial rates?
• RQ4 What conditions on the basis functions (e.g., trigonometric, Legendre, wavelets) ensure the validity of the asymptotic approximation?
• RQ5 Can the bias term in the projection estimator be controlled uniformly over the domain to ensure the overall convergence rate?

Key findings

• The maximal deviation distribution of the projection estimator converges to the Gumbel distribution with polynomial rate, improving upon prior logarithmic rates.
• For Legendre polynomial bases, the convergence rate of the deviation distribution is shown to be polynomial via the construction of accompanying laws with explicit expansions.
• The bias term in the estimator is bounded by $ \breve{c} n^{(3\varkappa/2)-1} m^{1/2} $, which is negligible under appropriate choices of $ m $ and $ n $.
• The normalized maximal deviation satisfies $ \mathbb{P}\left\{ \sqrt{\frac{T}{m}} \sup_{x\in D} \frac{|\hat{s}_n(x) - \mathbb{E}\hat{s}_n(x)|}{\sqrt{s(x)}} \leq u_m \right\} \to e^{-2e^{-y}} $, confirming Gumbel convergence.
• The sequence of accompanying laws $ W_m $ is constructed such that $ W_m \to -2e^{-y}(1 + R(m)) $, with $ R(m) \to 0 $ polynomially fast.
• The proof establishes uniform convergence over compact sets, allowing the use of the Gumbel limit law with error bounds decaying as $ O(n^{-\lambda}) $.