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Limit Theorems for Random Processes

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Part of the Classics in Mathematics book series

Abstract

Let ℋ be a metric space with metric ϱ(x, y), 𝓑 be the σ-algebra of its Borel subsets, 𝓒be the space of all bounded continuous functions defined on ℋ with the norm \( {\left\| f \right\|_x} = \mathop{{\sup }}\limits_x \left| {f(x)} \right| \). A sequence of measures μ n defined on 𝓑 is called weakly convergent to measure μ if for any function f in 𝓒 the relation
$$ \mathop{{\lim }}\limits_{{n \to \infty }} \int {f(x)} {\mu_n}\left( {dx} \right) = \int {f(x)} \mu \left( {dx} \right) $$
is satisfied. The set M of measures {μ} defined on 𝓑 is called weakly compact if from any sequence of measures μn in M a weakly convergent subsequence can be extracted.

Keywords

Limit Theorem Random Process Marginal Distribution Weak Convergence Independent Random Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and MechanicsAcademy of Sciences of the Ukrainian SSRDonetskUSSR
  2. 2.Institute of MathematicsAcademy of Sciences of the Ukrainian SSRKievUSSR

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