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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
MathSciNet review: 0313476
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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.


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  • [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
  • [5] Th. Kaluza, Über die Koeffizienten reziproker Potenzreihen, Math. Z. 28 (1928), no. 1, 161–170 (German). MR 1544949, 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 Orstein 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
  • [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

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DOI: https://doi.org/10.1090/S0002-9939-1973-0313476-9
Article copyright: © Copyright 1973 American Mathematical Society