This paper provides a strong approximation, or coupling, theory for spot volatility estimators formed using high-frequency data. We show that the t-statistic process associated with the nonparametric spot volatility estimator can be strongly approximated by a growing-dimensional vector of independent variables defined as functions of Brownian increments. We use this coupling theory to study the uniform inference for the volatility process in an infill asymptotic setting. Specifically, we propose uniform confidence bands for spot volatility, beta, idiosyncratic variance processes, and their nonlinear transforms. The theory is also applied to address an open question concerning the inference of monotone nonsmooth integrated volatility functionals such as the occupation time and its quantiles.
"Volatility coupling." Ann. Statist. 49 (4) 1982 - 1998, August 2021. https://doi.org/10.1214/20-AOS2023