**5. Discussion**

As it has been already mentioned, Theorem 2 provides the possibility to construct asymptotic confidence intervals for the mean-square thresholding risk. For practical purposes, it is desirable to have guaranteed confidence intervals. These intervals could be constructed based on the estimates of the convergence rate in Theorem 2. The estimates should depend on the Lipschitz parameters and parameter *α*. Guaranteed confidence intervals would help to explain how the results of Theorems 2 and 3 affect the error estimation for the finite signal size. We therefore leave the problem of estimation of the rate of convergence and explicit numerical simulation for future work.

The obtained results are applicable to Meyer wavelets. Their advantage is that they possess infinitely many vanishing moments. It simplifies the proof of asymptotic normality in [12]. In view of the results of [8] it is clear that similar conclusions could be obtained with other wavelets that have a large enough number of vanishing moments (e.g., various Daubechies families).

It follows from Theorems 2 and 3 that the statistical properties of the mean-square risk estimator in a model with the uniform random design remain the same as in a model with equispaced samples. Note that this situation is not common. Random times of sample registration can also result in a random sample size. This situation was considered in [22]. In this case, the properties of the model can significantly differ from the properties of the fixed sample size model. For example, the limit distribution of the mean-square risk estimator can be a scale mixture of normal laws, which can have

significantly heavier tails than the normal distribution. In particular, this distribution may belong to the class of stable laws, and it is well known that the variances of all stable laws, except the normal one, are infinite (the properties of stable distributions are discussed in detail in the monograph of V. M. Zolotarev [23]; see also [24]).

**Funding:** This research was funded by Russian Science Foundation, project number 18-11-00155.

**Conflicts of Interest:** The author declares no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
