Asymptotic properties of Markoff transition prababilities

Author:
J. L. Doob

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

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References | Similar Articles | Additional Information

**[1]**W. Ambrose, P. Halmos, and S. Kakutani,*The decomposition of measures*, II, Duke Math. J. vol. 9 (1942) pp. 43-47. MR**0005801 (3:210g)****[2]**D. Blackwell,*Idempotent Markoff chains*, Ann. of Math. vol. 43 (1942) pp. 560-567. MR**0006632 (4:17b)****[3]**J. L. Doob,*Stochastic processes with an integral-valued parameter*, Trans. Amer. Math. Soc. vol. 44, (1938) pp. 87-150. MR**1501964****[4]**-,*The Brownian movement and stochastic equations*, Ann. of Math. vol. 43 (1942) pp. 351-369. MR**0006634 (4:17d)****[5]**E. Hopf,*Ergodentheorie*, Ergebnisse der Mathematik, vol. 5, no. 2.**[6]**S. Kakutani,*Ergodic theorems and the Markoff process with a stable distribution*, Proc. Imp. Acad. Tokyo vol. 16 (1940) pp. 49-54. MR**0002049 (1:343b)****[7]**K. Yosida and S. Kakutani,*Operator-theoretical treatment of Markoff's process and mean ergodic theorem*, Ann. of Math. vol. 42 (1941) pp. 188-228. MR**0003512 (2:230e)****[8]**K. Yosida,*The Markoff process with a stable distribution*, Proc. Imp. Acad. Tokyo vol. 16 (1940) pp. 43-48. MR**0002048 (1:343a)****[9]**N. Wiener,*The ergodic theorem*, Duke Math. J. vol. 5 (1939) pp. 1-18. MR**1546100****[10]***(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 is absolutely continuous with respect to a given self-reproducing distribution , with a positive continuous density, and proves that then . This is a special case of Theorem 5.

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DOI:
https://doi.org/10.1090/S0002-9947-1948-0025097-6

Article copyright:
© Copyright 1948
American Mathematical Society