Limit theorems for stochastic processes (Q2726250)

This is the translation of the first outstanding work written by the author [Teor. Veroyatn. Primen. 1, 289-319 (1956; Zbl 0074.33802)]. It is the work where A. V. Skorokhod proposed the method of a single probability space and introduced several topologies in the space of all functions without discontinuities of the second kind, and proved some limit theorems that were the deep generalization of the famous Donsker invariant principle. NEWLINENEWLINENEWLINEUp to 1956 the limit theorems of probability theory, which previously dealt primarily with the theory of summation of independent random variables, have been extended rather widely to the theory of stochastic processes. Among the works done in this connection, there are the articles by \textit{A. N. Kolmogorov} [Bull. Acad. Sci. URSS, VII. Ser. 1933, No. 3, 363-372 (1933; Zbl 0006.35802), Izv. Akad. Nauk SSSR, Otd. Mat. Estest. Nauk, VII. Ser. No. 7, 959-962 (1931; Zbl 0003.35702)], \textit{M. D. Donsker} [Mem. Am. Math. Soc. 6 (1951; Zbl 0042.37602)], \textit{I. Gikhman} [Kyiv. Derzh. Univ. Mat. Sb. 7-75 (1953)], \textit{Yu. V. Prokhorov} [Teor. Veroyatn. Primen. 1, 177-238 (1956; Zbl 0075.29001), Usp. Mat. Nauk 8, No. 3(55), 165-167 (1953)], \textit{A. V. Skorokhod} [Dokl. Akad. Nauk SSSR 106, 781-784 (1956; Zbl 0074.12703) and ibid. 104, 364-367 (1955; Zbl 0068.12302)], \textit{N. N. Chentsov} [Teor. Veroyatn. Primen. 1 (1956)]. A. N. Kolmogorov, M. Donsker, I. I. Gikhman and A. V. Skorokhod treat various more or less special important cases of this kind of limit theorems. Yu. V. Prokhorov indicates a general approach to limit theorems for stochastic processes, based on compactness criteria of measures in a complete separable metric space. The restriction to complete metric spaces is not very natural, since in very specific cases it becomes necessary to find in the trajectory space of the random process a complete metric satisfying definite conditions. This is not always possible, and even when it is, it is not always simple. Skorokhod's article suggests a new approach to limit theorems which can be used for many topological spaces in which in general no complete metric exists.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].