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An Ergodic Theorem for Multidimensional Superadditive Processes

Published online by Cambridge University Press:  20 November 2018

Doğan Çömez
Doğan Çömez*
Affiliation:
North Dakota State University, Fargo, North Dakota
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The ergodic theorem for multidimensional strongly subadditive processes relative to a semigroup induced by a measure preserving point transformation on X was proved by R. T. Smythe [18]. His results have been generalized by M. A. Akçoğlu and U. Krengel [4] to the continuous parameter case. The definition of superadditivity they used is stronger than Smythe's but weaker than strong superadditivity. R. Emilion and B. Hachem [10] extended this result to strongly superadditive processes relative to a semigroup generated by a pair of commuting Markovian operators which are also L∞ -contractions. The basic tool in the proof is a technique which may be referred to as “reduction of dimension“ and they used a version of it due to A. Brunei [6].

The purpose of this paper is to show that if F = {F(uv)}u >0 is a bounded strongly superadditive process with respect to a two-dimensional strongly continuous Markovian semigroup of operators on L 1, then u-2F(uu) converges a.e. as u → ∞.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 4 , 01 August 1985, pp. 612 - 634
Copyright
Copyright © Canadian Mathematical Society 1985

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