This paper is concerned with a central limit theorem for quadratic variation when observations come as exit times from a regular grid. We discuss the special case of a semimartingale with deterministic characteristics and finite activity jumps in detail and illustrate technical issues in more general situations.
"A note on central limit theorems for quadratic variation in case of endogenous observation times." Electron. J. Statist. 11 (1) 963 - 980, 2017. https://doi.org/10.1214/17-EJS1252