An integral inequality on C ([0,1]) and dispersion of OLS under near-integration

We obtain an inequality for the sample variance of a vector Brownian motion on [0,1] and an associated Ornstein-Uhlenbeck process. The result is applied to a regression involving near-integrated regressors, and establishes that in the limit the dispersion of the least squares estimator is greater in the near-integrated than in the integrated case. Our proof uses a quite general integral inequality, which appears to be new.