Geodesic
On the speed of convergence in the ergodic theorem for shift operators
Canadian journal of mathematics, Tome 77 (2025) no. 6, pp. 1919-1937

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DOI

Given a probability space $(X,\mu )$, a square integrable function f on such space and a (unilateral or bilateral) shift operator T, we prove under suitable assumptions that the ergodic means $N^{-1}\sum _{n=0}^{N-1} T^nf$ converge pointwise almost everywhere to zero with a speed of convergence which, up to a small logarithmic transgression, is essentially of the order of $N^{-1/2}$. We also provide a few applications of our results, especially in the case of shifts associated with toral endomorphisms.
DOI : 10.4153/S0008414X24000658
Mots-clés : Ergodic theorem, convergence rate, shift operators, toral endomorphisms 
Chalmoukis, Nikolaos; Colzani, Leonardo; Gariboldi, Bianca; Monguzzi, Alessandro. On the speed of convergence in the ergodic theorem for shift operators. Canadian journal of mathematics, Tome 77 (2025) no. 6, pp. 1919-1937. doi: 10.4153/S0008414X24000658
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     title = {On the speed of convergence in the ergodic theorem for shift operators},
     journal = {Canadian journal of mathematics},
     pages = {1919--1937},
     year = {2025},
     volume = {77},
     number = {6},
     doi = {10.4153/S0008414X24000658},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X24000658/}
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