On convex power series of a conservative Markov operator
Authors:
S. R. Foguel and B. Weiss
Journal:
Proc. Amer. Math. Soc. 38 (1973), 325-330
MSC:
Primary 28A65
DOI:
https://doi.org/10.1090/S0002-9939-1973-0313476-9
MathSciNet review:
0313476
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: A. Brunel proved that a conservative Markov operator, , has a finite invariant measure if and only if every operator
where
and
is conservative.
In this note we give a different proof and study related problems.
- [1] A. Brunel, New conditions for existence of invariant measures in ergodic theory., Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970) Springer, Berlin, 1970, pp. 7–17. MR 0268355
- [2] William Feller, An introduction to probability theory and its applications. Vol. I, John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1957. 2nd ed. MR 0088081
- [3] Shaul R. Foguel, The ergodic theory of Markov processes, Van Nostrand Mathematical Studies, No. 21, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0261686
- [4] S. R. Foguel, Remarks on conservative Markov processes, Israel J. Math. 6 (1968), 381–383 (1969). MR 0243618, https://doi.org/10.1007/BF02771218
- [5] Th. Kaluza, Über die Koeffizienten reziproker Potenzreihen, Math. Z. 28 (1928), no. 1, 161–170 (German). MR 1544949, https://doi.org/10.1007/BF01181155
- [6] David G. Kendall, Renewal sequences and their arithmetic, Symposium on Probability Methods in Analysis (Loutraki, 1966) Springer, Berlin, 1967, pp. 147–175. MR 0224175
- [7] Shu-teh C. Moy, Period of an irreducible positive operator, Illinois J. Math. 11 (1967), 24–39. MR 0211470
- [8] Donald Ornstein and Louis Sucheston, An operator theorem on 𝐿₁ convergence to zero with applications to Markov kernels, Ann. Math. Statist. 41 (1970), 1631–1639. MR 0272057, https://doi.org/10.1214/aoms/1177696806
- [9] Frank Spitzer, Principles of random walk, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1964. MR 0171290
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28A65
Retrieve articles in all journals with MSC: 28A65
Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1973-0313476-9
Article copyright:
© Copyright 1973
American Mathematical Society