  The autocovariance and cross-covariance functions naturally appear in many
time series procedures (e.g., autoregression or prediction). Under assumptions,
empirical versions of the autocovariance and cross-covariance are
asymptotically normal with covariance structure depending on the second and
fourth order spectra. Under non-restrictive assumptions, we derive a bound for
the Wasserstein distance of the finite sample distribution of the estimator of
the autocovariance and cross-covariance to the Gaussian limit. An error of
approximation to the second-order moments of the estimator and an $m$-dependent
approximation are the key ingredients in order to obtain the bound. As a worked
example, we discuss how to compute the bound for causal autoregressive
processes of order 1 with different distributions for the innovations. To
assess our result, we compare our bound to Wasserstein distances obtained via
simulation.
