Stability of decomposition into components of a discontinuous distribution function in uniform metric
The first part is devoted to the characterization of various concepts of convergence of d.f.'s of real valued r.v.'s. The main attention is paid to the uniform convergence (on \(\mathbb{R})\) of d.f.'s (i.e. to convergence in the metric \(\rho)\) and to the strong convergence. The second part provides conditions for strong precompactness of a family of d.f.'s. In the third section the extension of previous results to bounded increasing functions is given. The final part deals with \(\rho\)-convergence \(\rho (F_n, F) \to 0\) \((n \to \infty)\) of convolutions \(F_n = F_{n1} * F_{n2}\) when \(F\) has at least one jump. As a corollary necessary and sufficient conditions are found for strong convergence of d.f.'s of \(\xi_{n,1} + \cdots + \xi_{n, k_n} - A_n\) to the Poisson d.f. with parameter \(\lambda > 0\) (here \(\xi_{n,1}, \dots, \xi_{n, k_n}\) are independent r.v.'s, \(A_n = \text{const})\).
characterization of various concepts of convergence
scientific article; zbMATH DE number 846598
scientific article