### 50 years sets with positive reach -- a survey.

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Invariance principle in ${L}^{2}(0,1)$ is studied using signed random measures. This approach to the problem uses an explicit isometry between ${L}^{2}(0,1)$ and a reproducing kernel Hilbert space giving a very convenient setting for the study of compactness and convergence of the sequence of Donsker functions. As an application, we prove a ${L}^{2}(0,1)$ version of the invariance principle in the case of $\varphi $-mixing random variables. Our result is not available in the $D(0,1)$-setting.

Random measures derived from a stationary process of compact subsets of the Euclidean space are introduced and the corresponding central limit theorem is formulated. The result does not require the Poisson assumption on the process. Approximate confidence intervals for the intensity of the corresponding random measure are constructed in the case of fibre processes.