Certain properties of triangular transformations of measures

• Date: 2008
• Main Author: Medvedev, K.V.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2008
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/4540
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Certain properties of triangular transformations of measures / K.V. Medvedev // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 95–99. — Бібліогр.: 12 назв.— англ.

We study the convergence of triangular mappings on R^n, i.e., mappings T such that the ith coordinate function Ti depends only on the variables x1, . . . ,xi. We show that, under broad assumptions, the inverse mapping to a canonical triangular transformation is canonical triangular as well. An example is constructed showing that the convergence in variation of measures is not sufficient for the convergence almost everywhere of the associated canonical triangular transformations. Finally, we show
 that the weak convergence of absolutely continuous convex measures to an absolutely continuous measure yields the convergence in variation. As a corollary, this implies the convergence in measure of the associated canonical triangular transformations.