Statistical error due to finite-time averaging

Previous results concerning statistical errors due to finite-time $T$ averaging in many experiments are made more precise. In particular, we show that the error does not always decrease according to a $T^{-\frac{1}{2}}$ law as $T\to \infty$. In the case of power or correlation function measurements, the departure from a $T^{-\frac{1}{2}}$ law may be due to an unbounded spectral density for some frequencies, which appears for some physical noises.