Asymptotic regularity of the wavelet methods of inverting linear homogeneous operators from observations recorded at random times

When solving inverse statistical problems, it is often necessary to invert some linear homogeneous operator and it is usually necessary to use regularization methods, since the observed data are noisy. Popular methods for noise suppression are the procedures of thresholding the expansion coefficients of the observed function. The advantages of these methods are their computational efficiency and the ability to adapt to both the type of operator and the local features of the estimated function. An analysis of the errors of these methods is an important practical task, since it allows one to evaluate the quality of both the methods themselves and the equipment used. Sometimes, the nature of the data is such that observations are recorded at random times. If the observation points form a variational series constructed from a sample of a uniform distribution on the data recording interval, then the use of conventional threshold processing procedures is adequate. The present author analyzes the estimate of the mean square risk in the problem of inversion of linear homogeneous operators and demonstrates that under certain conditions, this estimate is strongly consistent and asymptotically normal.