  Time series analysis is ubiquitous in many fields of science including
gravitational-wave astronomy, where strain time series are analyzed to infer
the nature of gravitational-wave sources, e.g., black holes and neutron stars.
It is common in gravitational-wave transient studies to apply a tapered window
function to reduce the effects of spectral artifacts from the sharp edges of
data segments. We show that the conventional analysis of tapered data fails to
take into account covariance between frequency bins, which arises for all
finite time series -- no matter the choice of window function. We discuss the
origin of this covariance and show that as the number of gravitational-wave
detections grows, and as we gain access to more high signal-to-noise ratio
events, this covariance will become a non-negligible source of systematic
error. We derive a framework that models the correlation induced by the window
function and demonstrate this solution using both data from the first
LIGO--Virgo transient catalog and simulated Gaussian noise.
