Geodesic
On the CLT for means under the rotation action. II
Teoriâ veroâtnostej i ee primeneniâ, Tome 51 (2006) no. 2, pp. 433-443 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We propose a method allowing us to build, for various typical means generated by the action of any given irrational rotation of the circle, examples of $L^2$ functions satisfying the central limit theorem (CLT). We consider for instance nonlinear means, and means along the sequence of squares. In the latter case, the circle method of Hardy–Littlewood is used. We also give an example of continuous Gaussian random Fourier series with sample paths satisfying both CLT and almost sure CLT.
Keywords: central limit theorem, almost sure central limit theorem, nonlinear averages, square averages, weighted averages, Gaussian randomization, random Fourier series, circle method.
Mots-clés : irrational rotations 
@article{TVP_2006_51_2_a14,
     author = {M. Weber},
     title = {On the {CLT} for means under the rotation {action.~II}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {433--443},
     year = {2006},
     volume = {51},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2006_51_2_a14/}
}
TY  - JOUR
AU  - M. Weber
TI  - On the CLT for means under the rotation action. II
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2006
SP  - 433
EP  - 443
VL  - 51
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TVP_2006_51_2_a14/
LA  - en
ID  - TVP_2006_51_2_a14
ER  - 
%0 Journal Article
%A M. Weber
%T On the CLT for means under the rotation action. II
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2006
%P 433-443
%V 51
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_2006_51_2_a14/
%G en
%F TVP_2006_51_2_a14
M. Weber. On the CLT for means under the rotation action. II. Teoriâ veroâtnostej i ee primeneniâ, Tome 51 (2006) no. 2, pp. 433-443. http://geodesic.mathdoc.fr/item/TVP_2006_51_2_a14/

[1] Berkes I., “On almost i.i.d. subsequences of the trigonometric system”, Lecture Notes in Math., 1332, 1988, 54–63 | MR | Zbl

[2] Bourgain J., “Pointwise ergodic theorems for arithmetic sets”, Inst. Hautes Études Sci. Publ. Math., no 69, 1989, 5–45 | MR

[3] Burton R., Denker M., “On the central limit theorem for dynamical systems”, Trans. Amer. Math. Soc., 302:2 (1987), 715–726 | DOI | MR | Zbl

[4] Denker M., “The central limit theorem for dynamical systems”, Banach Center Publ., 23 (1986), 33–62 | MR

[5] de la Rue T., Ladouceur S., Peskir G., Weber M., “On the central limit theorem for aperiodic dynamical systems and applications”, Teor. \u Imov\accent'26 ır. Mat. Stat., 57 (1997), 140–159 | MR | Zbl

[6] Giuliano Antonini R., Weber M., “The intersective ASCLT”, Stochastic Anal. Appl., 22:4 (2004), 1009–1025 | DOI | MR | Zbl

[7] Giuliano Antonini R., Weber M., “Counting occurrences in almost sure limit theorems”, Colloq. Math., 102:2 (2005), 271–290 | DOI | MR | Zbl

[8] Gordin M., Weber M., “On the almost sure central limit theorem for a class of $Z^d$-actions”, J. Theoret. Probab., 15:2 (2002), 477–501 | DOI | MR | Zbl

[9] Gordin M., Weber M., “A borderline Gaussian random Fourier series for the sampled convergence in variation”, J. Math. Anal. Appl., 318 (2006), 526–551 | DOI | MR | Zbl

[10] Kac M., “On the distribution of values of sums of the type $\sum f(2^kt)$”, Ann. Math., 47 (1946), 33–49 | DOI | MR | Zbl

[11] Kato Y., “Central limit theorem for Weyl automorphism”, Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs., 34:3 (1987), 1–10 | MR

[12] Lacey M., “On central limit theorems, modulus of continuity and Diophantine type of irrational rotations”, J. Anal. Math., 61 (1993), 47–59 | DOI | MR | Zbl

[13] Lacey M., Philipp W., “A Note on the almost sure central limit theorem”, Statist. Probab. Lett., 9 (1990), 201–205 | DOI | MR | Zbl

[14] Ledoux M., Talagrand M., Probability in Banach Spaces, Springer-Verlag, 1990, 480 pp. | MR | Zbl

[15] Mitrinovic D. S., Analytic Inequalities, Springer-Verlag, New York–Berlin, 1970, 400 pp. | MR | Zbl

[16] Olevskii A. M., Fourier Series with Respect to General Orthogonal Systems, Springer-Verlag, New York–Heidelberg, 1975, 136 pp. | MR | Zbl

[17] Philipp W., Stout W., Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc., 161, no 2, 1975, 1–140 | MR | Zbl

[18] Salem R., Zygmund A., “On lacunary trigonometric series”, Proc. Natl. Acad. Sci. USA, 33 (1947), 333–338 | DOI | MR | Zbl

[19] Volný D., “Invariance principles and Gaussian approximation for strictly stationary processes”, Trans. Amer. Math. Soc., 351:8 (1999), 3351–3371 | DOI | MR | Zbl

[20] Weber M., “Un théorème central limite presque sûr à moments généralisés pour les rotations irrationnelles”, Manuscripta Math., 101:2 (2000), 175–190 | DOI | MR | Zbl

[21] Weber M., Entropie métrique et convergence presque partout, Hermann, Paris, 1998, 151 pp. | MR | Zbl