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, , 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****[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