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A ratio limit theorem for contraction projections and applications

Published online by Cambridge University Press:  18 May 2009

P. E. Kopp
P. E. Kopp
Affiliation:
University of Hull
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The similarities between martingale convergence theory and pointwise ergodic theory are now well known [5, 7, 9, 10]. In [5] the similarity between the proofs of the Hopf– Dunford–Schwartz individual ergodic theorem and the martingale convergence theorem is systematically exploited to produce very general ” maximal ergodic ” inequalities for certain sequences of contractions on L1-spaces. A different approach by Rota [10] and Rao [9] leads to a unified convergence theory for martingales and Abel limits. Bishop [1] has produced ” upcrossing” inequalities which yield both theChacon-Ornstein theorem [4] and the martingale convergence theorem.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 14 , Issue 1 , February 1973 , pp. 80 - 85
Copyright
Copyright © Glasgow Mathematical Journal Trust 1973

References

REFERENCES

Bishop, E., Foundations of a constructive analysis (New York, 1967).Google Scholar
Douglas, R. G., Contractive projections on an L1-space, Pacific J Math. 15 (1965), 443–462.CrossRefGoogle Scholar
Dunford, N. and Schwartz, J. T., Linear operators Part I (New York, 1964).Google Scholar
Hopf, E., On the ergodic theorem for positive linear operators, j. Reine Angew. Math. 205 (1961), 101–106.Google Scholar
, A. and Tulcea, C. Ionescu, Abstract ergodic theorems, Trans Amer Math Soc. 107 (1963), 107–124.CrossRefGoogle Scholar
Ito, Y., Uniform integrability and the pointwise ergodic theorem, Proc. Amer. Math. Soc. 16 (1965), 222–227.CrossRefGoogle Scholar
Jerison, M., Martingale formulation of ergodic theorems, Proc. Amer. Math. Soc.. 10 (1959), 531–539.CrossRefGoogle Scholar
Kakutani, S., Concrete representations of abstract (L)-spaces and the mean ergodic theorem, Ann of Math. 42 (1941), 532–537.CrossRefGoogle Scholar
Rao, M. M., Generalised martingales, Contributions to probability and ergodic theory, Springer, Lecture Notes in Mathematics 160 (Berlin, 1970).Google Scholar
Rota, G. C., Une théorie unifiée des martingales et des moyennes ergodiques, C. R. Acad. Sci. Ser A 252 (1961), 2064–2066.Google Scholar
Wulbert, D. E., Characterisation of conditional expectation operators, Pacific J Math.. 34 (1970), 285–288.CrossRefGoogle Scholar