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Modified wavelet variation for the Hermite processes

Laurent Loosveldt, Ciprian A. Tudor

Abstract

We define an asymptotically normal wavelet-based strongly consistent estimator for the Hurst parameter of any Hermite processes. This estimator is obtained by considering a modified wavelet variation in which coefficients are wisely chosen to be, up to negligeable remainders, independent. We use Stein-Malliavin calculus to prove that this wavelet variation satisfies a multidimensional Central Limit Theorem, with an explicit bound for the Wasserstein distance.

Modified wavelet variation for the Hermite processes

Abstract

We define an asymptotically normal wavelet-based strongly consistent estimator for the Hurst parameter of any Hermite processes. This estimator is obtained by considering a modified wavelet variation in which coefficients are wisely chosen to be, up to negligeable remainders, independent. We use Stein-Malliavin calculus to prove that this wavelet variation satisfies a multidimensional Central Limit Theorem, with an explicit bound for the Wasserstein distance.
Paper Structure (9 sections, 18 theorems, 174 equations)

This paper contains 9 sections, 18 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. The wavelet coefficients and their decomposition
  3. The modified wavelet variation
  4. The special wavelet coefficients
  5. The modified wavelet variation and its negligible part
  6. The Central Limit Theorem for the modified wavelet variation
  7. Discretization of the wavelet variation
  8. Estimation of the Hurst parameter
  9. Appendix

Key Result

Lemma 1

Let $a,M>0$ and let $k$ be a positive integer. We have

Theorems & Definitions (19)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Proposition 1
  • Proposition 2
  • Definition 1
  • Lemma 6
  • Lemma 7
  • ...and 9 more