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Proceedings of the American Mathematical Society

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A geometric proof of Markov ergodic theorem


Author: R. Z. Yeh
Journal: Proc. Amer. Math. Soc. 26 (1970), 335-340
MSC: Primary 60.65
DOI: https://doi.org/10.1090/S0002-9939-1970-0263166-3
MathSciNet review: 0263166
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Abstract: A geometric approach combined with topological results leads to a criterion for ergodic stability of Markov transformations. The matrix representation of this criterion provides an alternative proof for the well-known theorem of Markov in probability.


References [Enhancements On Off] (What's this?)

  • [1] A. A. Markov, Investigation of a noteworthy case of dependent trials, Izv. Ros. Akad. Nauk 1 (1907) (Russian) or B. V. Gnedenko, Course in the theory of probability, Fizmatgiz, Moscow, 1961; English transl., Chelsea, New York, 1962, pp. 142-145. MR 25 #2622.
  • [2] V. Borovikov, On the intersection of a sequence of simplexes, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 6(52), 179–180 (Russian). MR 0053505
  • [3] F. R. Gantmacher, The theory of matrices, GITTL, Moscow, 1953; English transl., Vol. 2, Chelsea, New York, 1959, pp. 50-93. MR 16, 438; MR 21 #6372c.
  • [4] R. Z. Yeh, On the effect of an affine transformation on a certain $ k$-convex set, (to appear).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0263166-3
Keywords: Stochastic simplex, Markov transformation, sequence of convex sets, permutation of vertices, boundary, interior, fixed points, cyclic points, acyclic transformation, irreducible transformation, ergodic stability, transition matrix, hyperplane
Article copyright: © Copyright 1970 American Mathematical Society