We consider the problem of estimating the autocorrelation operator of an autoregressive Hilbertian process. By means of a Tikhonov approach, we establish a general result that yields the convergence rate of the estimated autocorrelation operator as a function of the rate of convergence of the estimated lag zero and lag one autocovariance operators. The result is general in that it can accommodate any consistent estimators of the lagged autocovariances. Consequently it can be applied to processes under any mode of observation: complete, discrete, sparse, and/or with measurement errors. An appealing feature is that the result does not require delicate spectral decay assumptions on the autocovariances but instead rests on natural source conditions. The result is illustrated by application to important special cases.

我们考虑估计自回归希尔伯特过程的自相关算子的问题。通过 Tikhonov 方法，我们建立了一个一般结果，该结果将估计自相关算子的收敛速度作为估计滞后零和滞后一自协方差算子的收敛速度的函数。结果是普遍的，因为它可以容纳滞后自协方差的任何一致估计量。因此，它可以应用于任何观察模式下的过程：完整的、离散的、稀疏的和/或有测量误差的。一个吸引人的特点是，结果不需要对自协方差进行精细的光谱衰减假设，而是依赖于自然源条件。通过应用到重要的特殊情况来说明结果。