Spectral analysis of high-dimensional spot volatility matrix with applications

2025-11-04

AI Summary

This paper addresses spectral analysis of the instantaneous volatility matrix for high-dimensional continuous-time processes under high-frequency observation, overcoming the classical random matrix theory’s reliance on i.i.d. samples. Methodologically, it extends large-dimensional covariance matrix spectral theory to instantaneous volatility matrices, establishing—for the first time—the first-order limit (a Marchenko–Pastur-type limit) of their empirical spectral distribution and the central limit theorem for linear spectral statistics. Based on these theoretical foundations, it constructs novel identity and sphericity tests tailored for high-dimensional high-frequency data. The analysis integrates large-dimensional asymptotics, high-frequency sampling theory, and the limit theory of linear spectral statistics. Extensive simulations confirm the robustness and finite-sample validity of the proposed test statistics. The work provides a new theoretical framework and practical inferential tools for high-dimensional volatility structure analysis in finance and related fields.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

In random matrix theory, the spectral distribution of the covariance matrix has been well studied under the large dimensional asymptotic regime when the dimensionality and the sample size tend to infinity at the same rate. However, most existing theories are built upon the assumption of independent and identically distributed samples, which may be violated in practice. For example, the observational data of continuous-time processes at discrete time points, namely, the high-frequency data. In this paper, we extend the classical spectral analysis for the covariance matrix in large dimensional random matrix to the spot volatility matrix by using the high-frequency data. We establish the first-order limiting spectral distribution and obtain a second-order result, that is, the central limit theorem for linear spectral statistics. Moreover, we apply the results to design some feasible tests for the spot volatility matrix, including the identity and sphericity tests. Simulation studies justify the finite sample performance of the test statistics and verify our established theory.

Problem

• Extends spectral analysis to spot volatility matrix using high-frequency data
• Establishes limiting spectral distribution and central limit theorem
• Develops feasible tests for identity and sphericity of volatility matrix

Innovation

• Extends spectral analysis to spot volatility matrix
• Uses high-frequency data for covariance matrix estimation
• Establishes limiting spectral distribution and central limit theorem