Electronic Journal of Probability

Central Limit Theorem for $\mathbb{Z}_{+}^d$-actions by toral endomorphisms

Mordechay Levin

Full-text: Open access

Abstract

In this paper we prove the central limit theorem  for the following multisequence $$\sum_{n_1=1}^{N_1} ... \sum_{n_d=1}^{N_d}   f(A_1^{n_1}...A_d^{n_d} {\bf x} )$$ where $f$ is a Hölder's continue function, $A_1,\ldots,A_d$ are $s\times s$ partially hyperbolic commuting  integer matrices, and $\bf x$ is a uniformly distributed random variable in $[0,1]^s$. Next we prove the functional central limit theorem, and the almost sure central limit theorem. The main tool is the $S$-unit theorem.

Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 35, 42 pp.

Dates
Accepted: 11 March 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465064260

Digital Object Identifier
doi:10.1214/EJP.v18-1904

Mathematical Reviews number (MathSciNet)
MR3035763

Zentralblatt MATH identifier
1306.60010

Subjects
Primary: 60F15: Strong theorems
Secondary: 37A

Keywords
Central limit theorem partially hyperbolic actions toral endomorphisms

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Levin, Mordechay. Central Limit Theorem for $\mathbb{Z}_{+}^d$-actions by toral endomorphisms. Electron. J. Probab. 18 (2013), paper no. 35, 42 pp. doi:10.1214/EJP.v18-1904. https://projecteuclid.org/euclid.ejp/1465064260


Export citation

References

  • Achieser, N. I. Theory of approximation. Translated by Charles J. Hyman Frederick Ungar Publishing Co., New York 1956 x+307 pp.
  • Alaca, Şaban; Williams, Kenneth S. Introductory algebraic number theory. Cambridge University Press, Cambridge, 2004. xviii+428 pp. ISBN: 0-521; 0-521-54011-9
  • Bary, N. K. A treatise on trigonometric series. Vols. I, II. Authorized translation by Margaret F. Mullins. A Pergamon Press Book The Macmillan Co., New York 1964 Vol. I: xxiii+553 pp. Vol. II: xix+508 pp.
  • Beck, József. Randomness in lattice point problems. Combinatorics, graph theory, algorithms and applications. Discrete Math. 229 (2001), no. 1-3, 29–55.
  • Berkes, István; Philipp, Walter; Tichy, Robert F. Empirical processes in probabilistic number theory: the LIL for the discrepancy of $(n_ k\omega)\bmod1$. Illinois J. Math. 50 (2006), no. 1-4, 107–145. ISBN: 0-9746986-1-X
  • Bickel, P. J.; Wichura, M. J. Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 (1971), 1656–1670.
  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9
  • Borevich, A. I.; Shafarevich, I. R. Number theory. Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20 Academic Press, New York-London 1966 x+435 pp.
  • Bulinski, Alexander; Shashkin, Alexey. Limit theorems for associated random fields and related systems. Advanced Series on Statistical Science & Applied Probability, 10. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. xii+436 pp. ISBN: 978-981-270-940-0; 981-270-940-1
  • Dick, Josef; Pillichshammer, Friedrich. Digital nets and sequences. Discrepancy theory and quasi-Monte Carlo integration. Cambridge University Press, Cambridge, 2010. xviii+600 pp. ISBN: 978-0-521-19159-3
  • Drmota, Michael; Tichy, Robert F. Sequences, discrepancies and applications. Lecture Notes in Mathematics, 1651. Springer-Verlag, Berlin, 1997. xiv+503 pp. ISBN: 3-540-62606-9
  • Einsiedler, Manfred; Ward, Thomas. Ergodic theory with a view towards number theory. Graduate Texts in Mathematics, 259. Springer-Verlag London, Ltd., London, 2011. xviii+481 pp. ISBN: 978-0-85729-020-5
  • Fazekas, István; Rychlik, ZdzisÅ‚aw. Almost sure central limit theorems for random fields. Math. Nachr. 259 (2003), 12–18.
  • Fortet, R. Sur une suite egalement répartie. (French) Studia Math. 9, (1940). 54–70.
  • Fukuyama, Katusi; Petit, Bernard. Le théorème limite central pour les suites de R. C. Baker. (French) [Central limit theorem for the sequences of R. C. Baker] Ergodic Theory Dynam. Systems 21 (2001), no. 2, 479–492.
  • Furstenberg, Harry. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 1967 1–49.
  • Gantmacher, F. R. The theory of matrices. Vols. 1, 2. Translated by K. A. Hirsch Chelsea Publishing Co., New York 1959 Vol. 1, x+374 pp. Vol. 2, ix+276 pp.
  • Hoffman, Kenneth; Kunze, Ray. Linear algebra. Second edition Prentice-Hall, Inc., Englewood Cliffs, N.J. 1971 viii+407 pp.
  • Hughes, C. P.; Rudnick, Z. On the distribution of lattice points in thin annuli. Int. Math. Res. Not. 2004, no. 13, 637–658.
  • Kac, M. On the distribution of values of sums of the type $\sum f(2^ k t)$. Ann. of Math. (2) 47, (1946). 33–49.
  • Katok, Anatole; Katok, Svetlana. Higher cohomology for abelian groups of toral automorphisms. II. The partially hyperbolic case, and corrigendum. Ergodic Theory Dynam. Systems 25 (2005), no. 6, 1909–1917.
  • Katok, Anatole; Niţică, Viorel. Rigidity in higher rank abelian group actions. Volume I. Introduction and cocycle problem. Cambridge Tracts in Mathematics, 185. Cambridge University Press, Cambridge, 2011. vi+313 pp. ISBN: 978-0-521-87909-5
  • Lang, Serge. Algebra. Revised third edition. Graduate Texts in Mathematics, 211. Springer-Verlag, New York, 2002. xvi+914 pp. ISBN: 0-387-95385-X
  • Le Borgne, Stéphane. Limit theorems for non-hyperbolic automorphisms of the torus. Israel J. Math. 109 (1999), 61–73.
  • Leonov, V. P. On the central limit theorem for ergodic endomorphisms of compact commutative groups. (Russian) Dokl. Akad. Nauk SSSR 135 1960 258–261.
  • Leonov, V. P. Nekotorye primeneniya starshikh semiinvariantov k teorii statsionarnykh sluchaĭnykh protsessov. (Russian) [Some applications of higher semi-invariants to the theory of stationary random processes] Izdat. “Nauka”, Moscow 1964 67 pp.
  • Levin, M. B., The multidimensional generalization of J.Beck 'Randomness of n sqrt 2 ; mod 1...' and a.s. invariance principle for ZZ^d-actions of toral automorphisms, Abstracts of Annual Meeting of the Israel Mathematical Union,(2002), http://imu.org.il/Meetings/IMUmeeting2002/ergodic.txt.
  • Levin, Mordechay B. On low discrepancy sequences and low discrepancy ergodic transformations of the multidimensional unit cube. Israel J. Math. 178 (2010), 61–106.
  • Levin, Mordechay B. Adelic constructions of low discrepancy sequences. Online J. Anal. Comb. No. 5 (2010), 27 pp.
  • Levin, M.B., A multiparameter variant of the Salem-Zygmund central limit theorem on lacunary trigonometric series, Colloq. Mathem., 17 p., to appear.
  • Levin, M.B., On Gaussian limiting distribution of lattice points in a parallelepiped, 45 p., submited.
  • Levin, Mordechay B.; Merzbach, Ely. Central limit theorems for the ergodic adding machine. Israel J. Math. 134 (2003), 61–92.
  • Lifshits, M. A. Almost sure limit theorem for martingales. Limit theorems in probability and statistics, Vol. II (Balatonlelle, 1999), 367–390, János Bolyai Math. Soc., Budapest, 2002.
  • Marcus, Marvin; Minc, Henryk. A survey of matrix theory and matrix inequalities. Reprint of the 1969 edition. Dover Publications, Inc., New York, 1992. xii+180 pp. ISBN: 0-486-67102-X
  • Miles, Richard; Ward, Thomas. A directional uniformity of periodic point distribution and mixing. Discrete Contin. Dyn. Syst. 30 (2011), no. 4, 1181–1189.
  • Móricz, F. A general moment inequality for the maximum of the rectangular partial sums of multiple series. Acta Math. Hungar. 41 (1983), no. 3-4, 337–346.
  • Noszály, Csaba; Tómács, Tibor. A general approach to strong laws of large numbers for fields of random variables. Ann. Univ. Sci. Budapest. Eōtvōs Sect. Math. 43 (2000), 61–78 (2001).
  • Philipp, Walter. Empirical distribution functions and strong approximation theorems for dependent random variables. A problem of Baker in probabilistic number theory. Trans. Amer. Math. Soc. 345 (1994), no. 2, 705–727.
  • Philipp, Walter; Stout, William. Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 2 (1975), issue 2, no. 161, iv+140 pp.
  • Schlickewei, H. P.; Schmidt, W. M. The number of solutions of polynomial-exponential equations. Compositio Math. 120 (2000), no. 2, 193–225.
  • Schmidt, Klaus; Ward, Tom. Mixing automorphisms of compact groups and a theorem of Schlickewei. Invent. Math. 111 (1993), no. 1, 69–76.
  • Zygmund, A. Trigonometric series. 2nd ed. Vols. I, II. Cambridge University Press, New York 1959 Vol. I. xii+383 pp.; Vol. II. vii+354 pp.