Portenko, N. I. The development of I. I. Gikham’s idea concerning the methods for investigating local behavior of diffusion processes and their weakly convergent sequences. (English. Russian original) Zbl 0869.60071 Theory Probab. Math. Stat. 50, 7-22 (1995); translation from Teor. Jmovirn. Mat. Stat. 50, 6-21 (1994). In his ancient little known papers [Trans. Kiev State Univ. 16, 149-164 (1957) and Bull. Kiev State Univ., Ser Astr., Mat., Mekh. 1, 25-46 (1958)] I. I. Gikhman characterized the weak convergence of a sequence of stochastic processes to a diffusion process in terms of the number of their crossings of a given surface. The author gives a review of these papers and of the related papers of Ukrainian probabilists developing Gikhman’s idea. Reviewer: V.Mackevičius (Vilnius) MSC: 60J60 Diffusion processes 58J65 Diffusion processes and stochastic analysis on manifolds 60-02 Research exposition (monographs, survey articles) pertaining to probability theory Keywords:weak convergence; crossings of a given surface PDFBibTeX XMLCite \textit{N. I. Portenko}, Theory Probab. Math. Stat. 50, 1 (1994; Zbl 0869.60071); translation from Teor. Jmovirn. Mat. Stat. 50, 6--21 (1994)