# Nonsynchronous covariation process and limit theorems

*Takaki Hayashi* and
*Nakahiro Yoshida*

*Stochastic Processes and their Applications*, 2011, vol. 121, issue 10, 2416-2454

**Abstract:**
An asymptotic distribution theory of the nonsynchronous covariation process for continuous semimartingales is presented. Two continuous semimartingales are sampled at stopping times in a nonsynchronous manner. Those sampling times possibly depend on the history of the stochastic processes and themselves. The nonsynchronous covariation process converges to the usual quadratic covariation of the semimartingales as the maximum size of the sampling intervals tends to zero. We deal with the case where the limiting variation process of the normalized approximation error is random and prove the convergence to mixed normality, or convergence to a conditional Gaussian martingale. A class of consistent estimators for the asymptotic variation process based on kernels is proposed, which will be useful for statistical applications to high-frequency data analysis in finance. As an illustrative example, a Poisson sampling scheme with random change point is discussed.

**Keywords:** Discrete; sampling; High-frequency; data; Martingale; central; limit; theorem; Nonsynchronicity; Quadratic; variation; Realized; volatility; Stable; convergence; Semimartingale (search for similar items in EconPapers)

**Date:** 2011

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