Convergence of the distribution of the threshold processing risk estimate to a mixture of normal laws at a random sample size

The popularity of signal processing algorithms using wavelet analysis methods has increased significantly over the past decades. This is explained by the fact that the wavelet decomposition is a convenient mathematical apparatus capable of solving problems in which the use of traditional Fourier analysis is ineffective. The main tasks for which the methods of wavelet analysis are used are signal compression and noise removal. In this case, the most commonly used method is threshold processing of wavelet expansion coefficients, which zeroes coefficients not exceeding a given threshold. The presence of noise and threshold processing procedures inevitably lead to errors in the estimated signal. The properties of estimates of such errors (mean square risk) have been studied in many papers. In particular, it has been shown that under certain conditions, the risk estimate is strongly consistent and asymptotically normal. When using threshold processing methods, it is usually assumed that the number of wavelet coefficients is fixed. However, in some situations, the sample size is not known in advance and is modeled by a random variable. In this paper, a model with a random number of observations is considered and a class of distributions is described that can be limiting for the mean-square risk estimate.