This work considers Gaussian process interpolation with a periodized version of the Mat{\'e}rn covariance function introduced by Stein (22, Section 6.7). Convergence rates are studied for the joint maximum likelihood estimation of the regularity and the amplitude parameters when the data is sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a fixed deterministic element of a Sobolev space of continuous functions is also considered, suggesting that bounding assumptions on some parameters can lead to different estimates.

翻译：本研究考虑使用Stein（22，第6.7节）引入的周期化Matérn协方差函数进行高斯过程插值。当数据按模型采样时，研究了联合最大似然估计正则化参数和振幅参数的收敛速率。同时分析了固定参数和估计参数下的均方积分误差，表明最大似然估计渐近产生的误差与已知真实参数时的误差相同。最后，还考虑了观测函数为索博列夫空间中连续函数的固定确定性元素的情况，表明对某些参数的边界假设可能导致不同的估计结果。