On convex power series of a conservative Markov operator
HTML articles powered by AMS MathViewer
- by S. R. Foguel and B. Weiss PDF
- Proc. Amer. Math. Soc. 38 (1973), 325-330 Request permission
Abstract:
A. Brunel proved that a conservative Markov operator, $P$, has a finite invariant measure if and only if every operator $Q = \Sigma _{n = 0}^\infty {\alpha _n}{P^n}$ where ${\alpha _n} \geqq 0$ and $\Sigma {\alpha _n} = 1$ is conservative. In this note we give a different proof and study related problems.References
- 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
- William Feller, An introduction to probability theory and its applications. Vol. I, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1957. 2nd ed. MR 0088081
- 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
- S. R. Foguel, Remarks on conservative Markov processes, Israel J. Math. 6 (1968), 381–383 (1969). MR 243618, DOI 10.1007/BF02771218
- Th. Kaluza, Über die Koeffizienten reziproker Potenzreihen, Math. Z. 28 (1928), no. 1, 161–170 (German). MR 1544949, DOI 10.1007/BF01181155
- David G. Kendall, Renewal sequences and their arithmetic, Symposium on Probability Methods in Analysis (Loutraki, 1966) Springer, Berlin, 1967, pp. 147–175. MR 0224175
- Shu-teh C. Moy, Period of an irreducible positive operator, Illinois J. Math. 11 (1967), 24–39. MR 211470
- Donald Ornstein and Louis Sucheston, An operator theorem on $L_{1}$ convergence to zero with applications to Markov kernels, Ann. Math. Statist. 41 (1970), 1631–1639. MR 272057, DOI 10.1214/aoms/1177696806
- 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
Additional Information
- © Copyright 1973 American Mathematical Society
- 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