, Volume 22, Issue 1, pp 97–101 | Cite as

Moving shift averages for ergodic transformations

  • E. Pfaffelhuber


The equality between the expectation value of a random variable and its shift average under an ergodic transformation is shown·to hold true if the end points of the interval along which the shifts are taken do not increase faster than a linear function of the interval length. If the increase is too fast, however, the moving shift average may, with probability 1, not converge to the expectation value.


Linear Function Stochastic Process Probability Theory Economic Theory Interval Length 
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  1. Erdös, P.: On a Theorem ofHsu andRobbins. Ann. Math. Stat.20, 286–291, 1949.Google Scholar
  2. Hsu, P.-L., andH. Robbins: Complete Convergence and the Law of Large Numbers. Proc. Nat. Acad. Sciences USA.33, No. 2, 25–31, 1947.Google Scholar

Copyright information

© Physica-Verlag Rudolf Liebing KG 1975

Authors and Affiliations

  • E. Pfaffelhuber
    • 1
  1. 1.Institut für InformationsverarbeitungUniversität TübingenTübingenGermany

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