We study the convergence of the ergodic averages where is a bounded sequence and a strictly increasing sequence of integers such that for some . Moreover we give explicit such sequences and and we investigate in particular the case where is a -multiplicative sequence.
Nous étudions la convergence de moyennes ergodiques où est une suite bornée et une suite d’entiers strictement croissante tels que avec . De plus nous donnons des exemples explicites de telles suites et puis, nous nous intéressons aux cas où est une suite - multiplicative.
Keywords: weighted ergodic averages, central limit theorem, almost sure convergence, -multiplicative sequences, substitutive sequences, generalized Thue-Morse sequences
@article{AIF_2002__52_2_561_0, author = {Durand, Fabien and Schneider, Dominique}, title = {Ergodic averages with deterministic weights}, journal = {Annales de l'Institut Fourier}, pages = {561--583}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {52}, number = {2}, year = {2002}, doi = {10.5802/aif.1894}, mrnumber = {1906483}, zbl = {1054.37003}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1894/} }
TY - JOUR TI - Ergodic averages with deterministic weights JO - Annales de l'Institut Fourier PY - 2002 DA - 2002/// SP - 561 EP - 583 VL - 52 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1894/ UR - https://www.ams.org/mathscinet-getitem?mr=1906483 UR - https://zbmath.org/?q=an%3A1054.37003 UR - https://doi.org/10.5802/aif.1894 DO - 10.5802/aif.1894 LA - en ID - AIF_2002__52_2_561_0 ER -
Durand, Fabien; Schneider, Dominique. Ergodic averages with deterministic weights. Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 561-583. doi : 10.5802/aif.1894. https://aif.centre-mersenne.org/articles/10.5802/aif.1894/
[AL] Generalized Rudin-Shapiro sequences, Acta Arith., Tome 60 (1991), pp. 1-27 | EuDML | MR | Zbl
[AM] On an extremal property of the Rudin-Shapiro sequence, Mathematika, Tome 32 (1985), pp. 33-38 | DOI | MR | Zbl
[Bo] Almost sure convergence and bounded entropy, Israel J. Math., Tome 63 (1988), pp. 79-97 | DOI | MR | Zbl
[DS] Théorèmes ergodiques aléatoires et suite de poids régularisants (2000) (Preprint LAMFA (Université de Picardie Jules Verne))
[G] Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith., Tome 13 (1967-1968), pp. 259-265 | EuDML | MR | Zbl
[GS] Ergodic theorems for dynamic random walks (1999) (preprint LAMFA (Université de Picardie Jules Verne)) | MR | Zbl
[JLO] Weighted ergodic theorems along subsequences of density zero, New York J. Math, Tome 3A (1997-1998), pp. 89-98 | EuDML | MR | Zbl
[Ka] Some random series of functions, Cambridge Studies in Advanced Mathematics (1985) | MR | Zbl
[Ke] Generalized Morse sequences, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Tome 10 (1968), pp. 335-353 | DOI | MR | Zbl
[KN] Uniform distribution of sequences, Wiley and Sons, 1974 | MR | Zbl
[Kr] Ergodic theorems, Studies in Mathematics, Tome 6, de Gruyter, Berlin-New York, 1985 | MR | Zbl
[La] On central limit theorems, modulus of continuity and Diophantine type for irrational rotations, J. Anal. Math, Tome 61 (1993), pp. 47-59 | DOI | MR | Zbl
[LM] Propriétés ergodiques des suites q-multiplicatives, Compositio Math., Tome 100 (1996), pp. 131-169 | Numdam | MR | Zbl
[LMM] Le théorème ergodique le long d'une suite $q$-multiplicative, Compositio Math., Tome 93 (1994), pp. 49-79 | Numdam | MR | Zbl
[M] Les suites à spectre vide et la répartition modulo 1, J. Number Theory, Tome 5 (1973), pp. 1-15 | DOI | MR | Zbl
[Mo] Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc, Tome 22 (1921), pp. 84-100 | DOI | JFM | MR
[MT] Dimension des courbes planes, papiers pliés et suites de Rudin-Shapiro, Bull. Soc. Math. France, Tome 109 (1981) no. 2, pp. 207-215 | Numdam | MR | Zbl
[Qu] Substitution dynamical systems--spectral analysis, Lecture Notes in Mathematics, Tome 1294, Springer-Verlag, Berlin, 1987 | MR | Zbl
[Ru] Some theorems on Fourier coefficients, Proc. Amer. Math. Soc, Tome 10 (1959), pp. 855-859 | DOI | MR | Zbl
[Sh] Extremal problems for polynomials and power series (1952) (Thesis, M.I.T)
[SW] Weighted averages of contractions along subsequences, Convergence in ergodic theory and probability (Columbus, OH, 1993), Tome 5 (1996), pp. 397-404 | Zbl
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