1956/01/01 by Yu. V. Prokhorov · 6 citations
Economics, Econometrics and Finance · Mathematics · #Mathematical Approximation and Integration #Stochastic processes and financial applications #advanced mathematical theories
paper · doi:10.1137/1101016
crossref issued 1956/01/01 · crossref published 1956/01/01 · crossref published-print 1956/01/01 · openalex publication_date 1956/01/01 · crossref created 2005/03/07 · crossref deposited 2017/01/29 · openalex created_date 2025/10/10 · crossref indexed 2026/07/31 · openalex updated_date 2026/08/01
The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.). Chapter 1. Let \Re be the c.s.m.s. and v a set of all finite measures on \Re . The distance L(μ 1 ,μ 2 ) (that is analogous to the Lévy distance) is introduced, and equivalence of L-convergence and w. c. is proved. It is shown that V\Re = (v,L) is c. s. m. s. Then, the necessary and sufficient conditions for compactness in V\Re are given. In section 1.6 the concept of “characteristic functionals” is applied to the study of w. cc of measures in Hilbert space. Chapter 2. On the basis of the above results the necessary and sufficient compactness conditions for families of probability measures in spaces C[0,1] and D[0,1] (space of functions that are continuous in [0,1] except for jumps) are formulated. Chapter 3. The general form of the “invariance principle” for the sums of independent random variables is developed. Chapter 4. An estimate of the remainder term in the well-known Kolmogorov theorem is given (cf. [3.1]).