@article{cdca4257c39b4a1ba38a37a8028eea9a,

title = "Nonsynchronous covariation process and limit theorems",

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, Semimartingale, Stable convergence",

author = "Takaki Hayashi and Nakahiro Yoshida",

note = "Funding Information: This work was supported in part by: (to Hayashi) the Grant-in-Aid for Scientific Research No. 19530186, Japan Society for the Promotion of Science (JSPS), The Japan Securities Scholarship Foundation, Grants for the Advancement of Research at the Graduate Schools of Keio University; (to Yoshida) JSPS Grants-in-Aid for Scientific Research No. 19340021, Cooperative Research Program of the Institute of Statistical Mathematics, Research program “BNP Paribas Fundamentals of Financial Mathematics”, JST Basic Research Programs PRESTO, and the Global COE program “The research and training center for new development in mathematics”, Graduate School of Mathematical Sciences, University of Tokyo; (to both) the 21st Century COE program “Base for New Development of Mathematics to Science and Technology”, Graduate School of Mathematical Sciences, University of Tokyo.",

year = "2011",

month = oct,

doi = "10.1016/j.spa.2010.12.005",

language = "English",

volume = "121",

pages = "2416--2454",

journal = "Stochastic Processes and their Applications",

issn = "0304-4149",

publisher = "Elsevier",

number = "10",

}