Asymptotic properties of Markoff transition prababilities

Author:
J. L. Doob

Journal:
Trans. Amer. Math. Soc. **63** (1948), 393-421

MSC:
Primary 60.0X

MathSciNet review:
0025097

Full-text PDF Free Access

References | Similar Articles | Additional Information

**[1]**Warren Ambrose, Paul R. Halmos, and Shizuo Kakutani,*The decomposition of measures. II*, Duke Math. J.**9**(1942), 43–47. MR**0005801****[2]**David Blackwell,*Idempotent Markoff chains*, Ann. of Math. (2)**43**(1942), 560–567. MR**0006632****[3]**J. L. Doob,*Stochastic processes with an integral-valued parameter*, Trans. Amer. Math. Soc.**44**(1938), no. 1, 87–150. MR**1501964**, 10.1090/S0002-9947-1938-1501964-2**[4]**J. L. Doob,*The Brownian movement and stochastic equations*, Ann. of Math. (2)**43**(1942), 351–369. MR**0006634****[5]**E. Hopf,*Ergodentheorie*, Ergebnisse der Mathematik, vol. 5, no. 2.**[6]**Shizuo Kakutani,*Ergodic theorems and the Markoff process with a stable distribution*, Proc. Imp. Acad. Tokyo**16**(1940), 49–54. MR**0002049****[7]**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**0003512****[8]**Kôsaku Yosida,*The Markoff process with a stable distribution*, Proc. Imp. Acad. Tokyo**16**(1940), 43–48. MR**0002048****[9]**Norbert Wiener,*The ergodic theorem*, Duke Math. J.**5**(1939), no. 1, 1–18. MR**1546100**, 10.1215/S0012-7094-39-00501-6**[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.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
60.0X

Retrieve articles in all journals with MSC: 60.0X

Additional Information

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

Article copyright:
© Copyright 1948
American Mathematical Society