Moderate deviations for bipower variation of general function and Hayashi-Yoshida estimators

TL;DR

This paper establishes moderate deviation principles for bipower variation and Hayashi-Yoshida estimators, advancing the probabilistic understanding of high-frequency financial data analysis.

Contribution

It introduces a new moderate deviations principle for m-dependent random variables, enhancing the theoretical framework for volatility estimators.

Findings

Derived moderate deviations for bipower variation

Established results for Hayashi-Yoshida estimator

Utilized Chen-Ledoux type condition for m-dependent variables

Abstract

We consider the moderate deviations behaviors for two (co-) volatility estima-tors: generalised bipower variation, Hayashi-Yoshida estimator. The results are obtained by using a new result about the moderate deviations principle for m-dependent random variables based on the Chen-Ledoux type condition. In the last decade there has been a considerable development of the asymptotic theory for processes observed at a high frequency. This was mainly motivated by financial applications , where the data, such as stock prices or currencies, are observed very frequently. As under the no-arbitrage assumptions price processes must follow a semimartingale, there was a need for probabilistic tools for functionals of semimartingales based on high frequency observations. Inspired by potential applications, probabilists started to develop limit theorems for semimartingales. Statisticians applied the asymptotic theory to analyze the path properties of discretely observed semimartingales: for the estimation of certain volatility functionals and realised jumps, or for performing various test procedures. We consider X t = (X 1,t , X 2,t) t$\in$[0,T ] a 2-dimensional semimartingale, defined on the filtred probability space ($\Omega$, F , (F t) [0,T ] , P), of the form