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Local asymptotic normality for mixed fractional Ornstein-Uhlenbeck process under high-frequency observation

Chunhao Cai, Yiwu Shang, Cong Zhang

Abstract

This paper consider the LAN property for the mixed O-U process under high-frequency observation when H>3/4. As considered in mixed fractional Brownian motion, we will also use the projection step to get the non-diagonal rate matrix.

Local asymptotic normality for mixed fractional Ornstein-Uhlenbeck process under high-frequency observation

Abstract

This paper consider the LAN property for the mixed O-U process under high-frequency observation when H>3/4. As considered in mixed fractional Brownian motion, we will also use the projection step to get the non-diagonal rate matrix.
Paper Structure (22 sections, 35 theorems, 345 equations, 2 figures)

This paper contains 22 sections, 35 theorems, 345 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation.
  3. Main Results
  4. The three basic matrices
  5. Preliminaries: A CLT for Gaussian quadratic forms
  6. Analysis of the score functions
  7. The sigma-direction of the score functions
  8. The H-direction of the score functions: removing the explicit logarithmic term
  9. Projection of the H-remainder against the sigma-direction
  10. The alpha-direction of the score functions and the second projection
  11. Joint diagonalized central sequence
  12. The LAN property in high-frequency observation via matrix expansions
  13. Matrix expansion of the likelihood ratio
  14. Figure design for the three dimensional case
  15. Concluding summary
  16. ...and 7 more sections

Key Result

Theorem 2.1

Assume $H\in(3/4,1)$, $\sigma>0$, $\alpha>0$, and $\Delta_n\to0$. For the projected score CLTs we impose and for the final LAN remainder control we further assume the polynomial mesh condition Let $M_n$ be the lower-triangular score transformation defined in eq:Mn-def, and let For every fixed $h\in\mathbb{R}^3$, set $\theta_{n,h}=\theta+r_n(\theta)^{-1}h$. Then where The definitions of the sy

Figures (2)

  • Figure 1: Schematic low frequency profile. The information is concentrated on the scale $\lambda\asymp\Delta_n$, hence after the rescaling $\lambda=\Delta_nu$ the effective weight becomes $w(u)$.
  • Figure 2: Pairwise contour plots for the projected Gaussian limit. Because the limiting covariance matrix is diagonal in the projected coordinates, the contours are axis-aligned. This is the most informative visual summary in dimension three.

Theorems & Definitions (76)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • ...and 66 more