Ergodicity bounds in the Sliced Wasserstein distance for Schur stable autoregressive processes

Abstract

Explicit calculations in dimension one show for Schur stable autoregressive processes with standard Gaussian noise that the ergodic convergence in the Wasserstein-$2$ distance is essentially given by the sum of the mean, which decays exponentially, and the standard deviation, which decays with twice the speed. This paper starts by showing new upper and lower multivariate affine transport bounds for the Wasserstein-$r$ distance for $r$ greater and equal to $1$. These bounds allow to formulate a novel sufficient (non-Gaussian) ergodic interpolation condition for the mentioned mean-variance behavior to take place in case of more general Schur stable multivariate autoregressive processes. All ergodic estimates are non-asymptotic with completely explicit constants. The main applications are precise thermalization bounds for Schur stable $\mathsf{AR}(p)$ and $\mathsf{ARMA}(p,q)$ models in Wasserstein and Sliced Wasserstein distance. In the sequel we establish with the help of coupling techniques explicit upper and lower exponential bounds for more general multivariate Schur stable autoregressive processes. This includes parallel sampling and the convergence of the empiricial means. The utility of our results in particular for the Sliced Wasserstein distance are confirmed by multivariate numerical experiments.