Strong consistency of the mean square risk estimate in the inverse statistical problems

Nonlinear methods of digital signal processing based on thresholding of wavelet coefficients became a popular tool for solving the problems of signal de-noising and compression. This is explained by the fact that the wavelet methods allow much more effective analysis of nonstationary signals than traditional Fourier analysis, thanks to the better adaptation to the functions with varying degrees of regularity. Wavelet thresholding risk analysis is an important practical task, because it allows determining the quality of techniques themselves and the equipment which is being used. In some applications, the data are observed not directly but after applying a linear transformation. The problem of inverting this transformation is usually set incorrectly, leading to an increase in the noise variance. In this paper, the asymptotic properties of the mean square error (MSE) estimate are studied when inverting linear homogeneous operators by means of wavelet vaguelette decomposition and thresholding. The strong consistency of this estimate has been proved under mild conditions.