Abstract
A Markov transition probability function was shown by Harris [3] to admit a σ-finite invariant measure, under a certain probabilistic recurrence condition which was later given an essentially equivalent measure-theoretic form (see [2], [4], [5], and [8]). The latter includes the assumption that if T is the L∞ operator induced by the system, then the L∞ norm of T is one. The present paper weakens this assumption replacing it by: lim inf Tn h < ∞ a.e.
Keywords
- Invariant Measure
- Countable Union
- Positive Linear Operator
- Small Function
- Markov Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Research supported by the National Science Foundation, Grants GP 8781 and GP 7693.
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References
J. L. Doob, Stochastic Processes, New York, Wiley and Sons, 1953.
J. Feldman, "Integral kernels and invariant measures for Markov transition functions," Ann. Math. Stat., 36, (1965), 517–523.
T. E. Harris, "The existence of stationary measures for certain Markov processes," Proc. Third Berkeley Sumposium on Mathematical Statistics II (1956), 113–124.
R. Isaac, "Non-singular Markov processes have stationary measures," Ann. Math. Stat., 35, (1964), 869–871.
N. C. Jain, "A note on invariant measures," Ann. Math. Stat., 37, (1966), 729–732.
N. Jain and B. Jamison, "Contributions to the Doeblin's theory of Markov processes," Z. Wahrscheinlichkeitstheorie verw. Geb. 8, (1967), 19–40.
B. Jamison and S. Orey, "Markov chains recurrent in the sense of Harris," Z. Wahrscheinlichkeitstheorie verw. Geb. 8, (1967), 41–48.
M. Metivier, "Existence of an invariant measure and an Ornstein's ergodic theorem," Ann. Math. Stat., 40, (1969), 79–96.
S. T. C. Moy, "The continuous part of a Markov operator," J. Math. Mech., 18, (1968), 137–142.
J. Neveu, "Mathematical Foundations of the Calculus of Probability," Holden-Day, San Francisco, 1965.
D. S. Ornstein, "The sums of iterates of a positive operator," To appear.
D. S. Ornstein and L. Sucheston, "An operator theorem on L1 convergence to zero, with applications to Markov kernels," Ann. Math. Stat., to appear.
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© 1970 Springer-Verleg
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Ornstein, D.S., Sucheston, L. (1970). On the existence of a σ-finite invariant measure under a generalized Harris condition. In: Contributions to Ergodic Theory and Probability. Lecture Notes in Mathematics, vol 160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0060655
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DOI: https://doi.org/10.1007/BFb0060655
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