# Minimax rates for the covariance estimation of multi-dimensional L\'evy processes with high-frequency data

## TL;DR

This paper develops a spectral estimator for co-integrated volatility in multi-dimensional Lévy processes using high-frequency data, establishing minimax convergence rates and comparing efficiency with existing methods.

## Contribution

It introduces a new spectral estimator for co-integrated volatility and proves its minimax optimality for a broad class of Lévy processes.

## Key findings

- Convergence rates are 1/√n for r ≤ 1 and (n log n)^{(r-2)/2} for r > 1.
- The estimator is minimax optimal within the specified class.
- The method effectively bounds co-jump activity using the harmonic mean.

## Abstract

This article studies nonparametric methods to estimate the co-integrated volatility for multi-dimensional L\'evy processes with high frequency data. We construct a spectral estimator for the co-integrated volatility and prove minimax rates for an appropriate bounded nonparametric class of semimartingales. Given $ n $ observations of increments over intervals of length $1/n$, the rates of convergence are $1 / \sqrt{n} $ if $ r \leq 1 $ and $ (n\log n)^{(r-2)/2} $ if $ r>1 $, which are optimal in a minimax sense. We bound the co-jump index activity from below with the harmonic mean. Finally, we assess the efficiency of our estimator by comparing it with estimators in the existing literature.