Wasserstein distance bounds on the normal approximation of empirical autocovariances and cross-covariances under non-stationarity and stationarity

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 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.

Automated summary

The autocovariance and cross-covariance functions are important in time series procedures and their empirical versions can be asymptotically normal under certain assumptions. In this study, we derive a bound for the Wasserstein distance between the finite-sample distribution of the estimator and the Gaussian limit. We show that an approximation error and an m-dependent approximation play key roles in obtaining the bound. We also provide an example of computing the bound for causal autoregressive processes with different innovation distributions and compare it to simulations using Wasserstein distances.