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Proceedings of the American Mathematical Society

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An ergodic theorem for Fréchet-valued random variables

Authors: D. Landers and L. Rogge
Journal: Proc. Amer. Math. Soc. 72 (1978), 49-53
MSC: Primary 60F15; Secondary 28A65
MathSciNet review: 0501293
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Abstract: We generalize the classical ergodic theorem from real-valued random variables to Fréchet space-valued random variables and obtain this generalization as a direct corollary of the classical theorem. As an application we obtain several strong laws of large numbers for Fréchet-valued random variables. In a similar way we obtain a martingale theorem for Fréchet-valued random variables.

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  • [1] Michel Métivier, Reelle und vektorwertige Quasimartingale und die Theorie der stochastischen Integration, Lecture Notes in Mathematics, Vol. 607, Springer-Verlag, Berlin-New York, 1977 (German). MR 0651570
  • [2] J. Neveu, Discrete-parameter martingales, Revised edition, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Translated from the French by T. P. Speed; North-Holland Mathematical Library, Vol. 10. MR 0402915
  • [3] W. J. Padgett and R. L. Taylor, Laws of large numbers for normed linear spaces and certain Fréchet spaces, Lecture Notes in Mathematics, Vol. 360, Springer-Verlag, Berlin-New York, 1973. MR 0426114
  • [4] F. W. Schäfke, Integrationstheorie. I, J. Reine Angew. Math. 244 (1970), 154–176 (German). MR 0271300
  • [5] F. W. Schäfke, Integrationstheorie. II, J. Reine Angew. Math. 248 (1971), 147–171 (German). MR 0285687
  • [6] F. W. Schäfke, Integrationstheorie und quasinormierte Gruppen, J. Reine Angew. Math. 253 (1972), 117–137 (German). MR 0306440
  • [7] R. L. Taylor and W. J. Padgett, Some laws of large numbers for normed linear spaces, Sankhyā Ser. A 36 (1974), no. 4, 359–368. MR 0385986

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Keywords: Pointwise ergodic theorem, strong law of large numbers, weakly orthogonal processes
Article copyright: © Copyright 1978 American Mathematical Society