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Measures in a functional spaces. Weak convergence, probability metrics. Functional limit theorems

  • Dmytro Gusak
  • Alexander Kukush
  • Alexey Kulik
  • Yuliya Mishura
  • Andrey Pilipenko
Chapter
Part of the Problem Books in Mathematics book series (PBM)

Abstract

In this chapter, we consider random elements taking values in metric spaces and their distributions. The definition of a random element taking values in \({{\Bbb X}}\) involves the predefined \(\sigma\)-algebra \({{\rm X}}\) of subsets of \({{\Bbb X}}\). The following statement shows that in a separable metric space, in fact, the unique natural choice for the \(\sigma\)-algebra \({{\rm X}}\) is the Borel \(\sigma\)-algebra \({{\rm B}}({{\Bbb X}})\).

Keywords

Random Walk Wiener Process Random Element Polish Space Invariance Principle 
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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Dmytro Gusak
    • 1
  • Alexander Kukush
    • 2
  • Alexey Kulik
    • 1
  • Yuliya Mishura
    • 3
  • Andrey Pilipenko
    • 1
  1. 1.Institute of Mathematics of Ukrainian National Academy of SciencesKyivUkraine
  2. 2.Department of Mathematical Analysis Faculty of Mechanics and MathematicsNational Taras Shevchenko University of KyivKyivUkraine
  3. 3.Department of Probability Theory and Mathematical Statistics Faculty of Mechanics and MathematicsNational Taras Shevchencko University of KyivKyivUkraine

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