Asymptotic properties of Markoff transition prababilities
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- by J. L. Doob
- Trans. Amer. Math. Soc. 63 (1948), 393-421
- DOI: https://doi.org/10.1090/S0002-9947-1948-0025097-6
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References
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- David Blackwell, Idempotent Markoff chains, Ann. of Math. (2) 43 (1942), 560–567. MR 6632, DOI 10.2307/1968811
- J. L. Doob, Stochastic processes with an integral-valued parameter, Trans. Amer. Math. Soc. 44 (1938), no. 1, 87–150. MR 1501964, DOI 10.1090/S0002-9947-1938-1501964-2
- J. L. Doob, The Brownian movement and stochastic equations, Ann. of Math. (2) 43 (1942), 351–369. MR 6634, DOI 10.2307/1968873 E. Hopf, Ergodentheorie, Ergebnisse der Mathematik, vol. 5, no. 2.
- Shizuo Kakutani, Ergodic theorems and the Markoff process with a stable distribution, Proc. Imp. Acad. Tokyo 16 (1940), 49–54. MR 2049
- Kôsaku Yosida and Shizuo Kakutani, Operator-theoretical treatment of Markoff’s process and mean ergodic theorem, Ann. of Math. (2) 42 (1941), 188–228. MR 3512, DOI 10.2307/1968993
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- Norbert Wiener, The ergodic theorem, Duke Math. J. 5 (1939), no. 1, 1–18. MR 1546100, DOI 10.1215/S0012-7094-39-00501-6 (Added in proof.) A. M. Yaglom, The ergodic principle for Markov processes with stationary distributions, C. R. (Doklady) Acad. Sci. URSS. N.S. vol. 54 (1947) pp. 347-349. The author supposes that ${P^{(s)}}(x,A)$ is absolutely continuous with respect to a given self-reproducing distribution $\Phi (A)$, with a positive continuous density, and proves that then ${\lim _{s \to \infty }}{P^{(s)}}(x,A) = \Phi (A)$. This is a special case of Theorem 5.
Bibliographic Information
- © Copyright 1948 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 63 (1948), 393-421
- MSC: Primary 60.0X
- DOI: https://doi.org/10.1090/S0002-9947-1948-0025097-6
- MathSciNet review: 0025097