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Moyennes uniformes et moyennes suivant une marche aléatoire
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  • Published: November 1988

Moyennes uniformes et moyennes suivant une marche aléatoire

  • Jean-Pierre Kahane1,
  • Jacques Peyrière1,
  • Wen Zhi-ying2 &
  • …
  • Wu Li-ming2 

Probability Theory and Related Fields volume 79, pages 625–628 (1988)Cite this article

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  • 4 Citations

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Summary

Let ϕ be a bounded function on ℤ such that \(\frac{{\text{1}}}{n}\sum\limits_{j = 1}^n {\varphi {\text{(}}m - j{\text{)}}} \) converges towards l as n goes to infinity, uniformly with respect to m. Let {X n} be a random walk on ℤ, not concentrated on a proper subgroup of ℤ Then, with probability 1, \(\frac{{\text{1}}}{n}\sum\limits_{j = 1}^n {\varphi {\text{(}}X_j {\text{)}}} \) converges towards l as n goes to infinity. The result also holds for any countable abelian group instead of ℤ. Other modes of convergence are considered (Cesaro convergence of order α>1/2). The Cesaro convergence of expressions such that ϕ(X n) ψ (X n+1) is also investigated.

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References

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Author information

Authors and Affiliations

  1. Centre D'Orsay, Université de Paris-Sud, Mathematique, Bâtiment 425, I-91405, Orsay Cedex, France

    Jean-Pierre Kahane & Jacques Peyrière

  2. Department of Mathematics, University of Wuhan, Wuhan, People's Republic of China

    Wen Zhi-ying & Wu Li-ming

Authors
  1. Jean-Pierre Kahane
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  2. Jacques Peyrière
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  3. Wen Zhi-ying
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  4. Wu Li-ming
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Kahane, JP., Peyrière, J., Zhi-ying, W. et al. Moyennes uniformes et moyennes suivant une marche aléatoire. Probab. Th. Rel. Fields 79, 625–628 (1988). https://doi.org/10.1007/BF00318786

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  • Received: 09 April 1987

  • Revised: 12 March 1988

  • Issue Date: November 1988

  • DOI: https://doi.org/10.1007/BF00318786

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