标题： Sliced Wasserstein距离中Schur稳定的自回归过程的遍历性界限 显示英文标题

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

摘要： 显式计算在维度一中表明，对于具有标准高斯噪声的Schur稳定自回归过程，Wasserstein-$2$距离中的遍历收敛基本上由均值的和给出，该均值呈指数衰减，以及标准差，其衰减速度是两倍。 本文首先展示了Wasserstein-$r$距离的新的上下界多变量仿射传输边界，适用于$r$大于等于$1$的情况。 这些界限允许为所述均值-方差行为在更一般的Schur稳定多变量自回归过程中发生的情况，制定一个充分（非高斯）的仿射-遍历-插值条件。 所有遍历估计都是非渐近的，并且具有完全明确的常数。 主要应用是在Wasserstein和切片Wasserstein距离下，对Schur稳定的$\textsf{AR}(p)$和$\textsf{ARMA}(p,q)$模型的精确热化界限。 随后，我们借助耦合技术，为更一般的多变量Schur稳定自回归过程建立了显式的上下界指数界限。 这包括并行采样和经验均值的收敛性。 显示英文摘要

摘要： 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 sufficient (non-Gaussian) affine-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 $\textsf{AR}(p)$ and $\textsf{ARMA}(p,q)$ 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.