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：本文研究基于Stein (22, 第6.7节)引入的周期化Matérn协方差函数的高斯过程插值。当数据按模型采样时，研究了正则性参数与振幅参数的联合最大似然估计的收敛速度。分析了固定参数与估计参数下的平均积分平方误差，表明最大似然估计在渐近意义上产生的误差与已知真实参数时相同。最后，考虑观测函数为连续函数Sobolev空间中的固定确定性元素的情况，表明对某些参数施加有界假设可能导致不同的估计结果。