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Markov chains: Discrete and continuous time

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

Abstract

Let phase space \(\Bbb{X}\) of a random sequence \(\{X_n,\,n\in\Bbb{Z}^+\}\) be enumerable. The sequence \(\{X_n,\,n\in\Bbb{Z}^+\}\) is called a Markov chain if
$$\forall n\in\Bbb{N}\ \forall i_1,\ldots,i_n,i_{n+1}\in \Bbb{X} \ \forall t_1\leq\cdots\leq t_n\leq t_{n+1}\in\Bbb{Z}_+:$$
$$\mathsf{P}(X_{t_{n+1}}=i_{n+1} / X_{t_{1}}=i_{1},\ldots,X_{t_{n}}=i_{n})=\mathsf{P}(X_{t_{n+1}}=i_{n+1} / X_{t_{n}}=i_{n}).$$

Keywords

Phase Space Markov Chain Random Walk Markov Process Stationary Distribution 
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.

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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