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