Summary
Let θ be an algebraic number greater than 1 andf a real 1-periodic function; ifF N denotes the random variable defined on [0, 1] byF N (t)\( = \frac{1}{{\sqrt N }}\sum\limits_0^{N - 1} {f(\theta ^n t)} \), it is proved here that under sufficiently broad assumptions onf: 1) the sequence\(\left\{ {\mathop \smallint \limits_0^1 F_N^2 (t) dt} \right\}\) converges to a finite σ2(σ≧0); 2) if σ>0, the sequence {F N } converges in law to\(\mathfrak{N}(0,\sigma ^2 )\). We give an explicit computation of σ with respect to θ and a characterisation of functions for which σ=0.
(Our results are also valid for almost every real θ>1).
Références
Bourgain, J.: The Riesz-Raikov theorem for algebraic numbers (a paraître)
Crépel, P.: Variables aléatoires dépendantes. In: Outils et modèles mathématiques pour l'automatique. Paris: Editions du C.N.R.S. 1981
Fortet, R.: Sur une suite également répartie. Stud. Math.9, 54–70 (1940)
Ibragimov, I.A.: The central limit theorem for sums of independent variables and sums of the form Σf(2k t). Theory Probab. Appl.XII (4), 596–607 (1967)
Kac, M.: On the distribution of values of sums of the type Σf(2k t). Ann. Math.47(1), 33–49 (1946)
Keller, G.: Un théorème de la limite centrale pour une classe de transformations monotones par morceaux. C.R. Acad. Sci. Paris, Ser. A291, 155–158 (1980)
Koksma, J.F.: Diophantische Approximationen. Berlin Heidelberg New York: Springer 1936
Komlos, J.: A central limit theorem for multiplicative systems. Canad. Math. Bull.16(1), 67–73 (1973)
Le Veque, W.J.: Reviews in Number Theory. Providence, RI: Am. Math. Soc. 1974
Maruyama, G.: On an asymptotic property of a gap sequence. Kôdai Math. Sem. Rep.1950, 31–32 (1950)
Rauzy, G.: Propriétés statistiques de suites arithmétiques. Paris: P.U.F. 1976
Salem, R., Zygmund, A.: On lacunary trigonometric series I/II. Proc. Natl. Acad. USA33/34, 333–338 /54–62 (1947/1948)
Takahashi, S.: On the distribution of values of the type Σf(q k t). Tôhoku Math. J.14(4), 233–249 (1962)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Petit, B. Le théorème limite central pour des sommes de Riesz-Raikov. Probab. Th. Rel. Fields 93, 407–438 (1992). https://doi.org/10.1007/BF01192715
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01192715