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.

翻译：自协方差与互协方差函数自然出现在许多时间序列过程中（例如自回归或预测）。在假设条件下，自协方差与互协方差的经验估计量渐近服从正态分布，其协方差结构依赖于二阶和四阶谱。在非限制性假设下，我们推导了自协方差与互协方差估计量有限样本分布与高斯极限之间Wasserstein距离的界。估计量二阶矩的近似误差和$m$相依近似是获得该界的关键要素。作为实例分析，我们讨论了如何针对具有不同创新分布的一阶因果自回归过程计算该界。为评估结果，我们将该界与通过模拟获得的Wasserstein距离进行了比较。