The central limit theorem for Riesz–Raikov sums II
HTML articles powered by AMS MathViewer
- by Katusi Fukuyama PDF
- Trans. Amer. Math. Soc. 372 (2019), 1193-1211 Request permission
Abstract:
For a $d\times d$ expanding matrix $A$, we investigate randomness of the sequence $\{A^k \boldsymbol x\}$ and prove the central limit theorem for $\sum f(A^k \boldsymbol x)$, where $f$ is a periodic function with a mild regularity condition.References
- M. Csörgő and P. Révész, Strong approximations in probability and statistics, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 666546
- Jean-Pierre Conze, Stéphane Le Borgne, and Mikaël Roger, Central limit theorem for stationary products of toral automorphisms, Discrete Contin. Dyn. Syst. 32 (2012), no. 5, 1597–1626. MR 2871327, DOI 10.3934/dcds.2012.32.1597
- A.-H. Fan, Decay of correlation for expanding toral endomorphims, Dynamical Systems, Proceedings of the International Conference in Honor of Professor Liao Shantao, Peking University, Beijing, 1998, Y. P. Jiang and L. Wen (eds.), World Scientific, Singapore, 1999, pp. 29–40.
- Ai Hua Fan, Équirépartition des orbites d’un endomorphisme de $\textbf {R}^d$, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 11, 735–738 (French, with English summary). MR 1139828
- R. Fortet, Sur une suite egalement répartie, Studia Math. 9 (1940), 54–70 (French, with Ukrainian summary). MR 5546, DOI 10.4064/sm-9-1-54-70
- Katusi Fukuyama, The central limit theorem for Riesz-Raikov sums, Probab. Theory Related Fields 100 (1994), no. 1, 57–75. MR 1292190, DOI 10.1007/BF01204953
- Katusi Fukuyama and Noriyuki Kuri, The central limit theorem for complex Riesz-Raikov sums, C. R. Math. Acad. Sci. Paris 353 (2015), no. 8, 749–753 (English, with English and French summaries). MR 3367646, DOI 10.1016/j.crma.2015.04.020
- M. Kac, On the distribution of values of sums of the type $\sum f(2^k t)$, Ann. of Math. (2) 47 (1946), 33–49. MR 15548, DOI 10.2307/1969033
- J. Komlós and P. Révész, Remark to a paper of Gaposhkin, Acta Sci. Math. (Szeged) 33 (1972), 237–241. MR 320617
- V. P. Leonov, On the central limit theorem for ergodic endomorphisms of compact commutative groups, Dokl. Akad. Nauk SSSR 135 (1960), 258–261 (Russian). MR 0171302
- Emmanuel Lesigne, Loi des grands nombres pour des sommes de Riesz-Raikov multidimensionnelles, Compositio Math. 110 (1998), no. 1, 39–49 (French, with English summary). MR 1601658, DOI 10.1023/A:1000281522303
- Mordechay B. Levin, Central limit theorem for $\Bbb Z^d_+$-actions by toral endomorphisms, Electron. J. Probab. 18 (2013), no. 35, 42. MR 3035763, DOI 10.1214/EJP.v18-1904
- T. Löbbe, Limit theorems for multivariate lacunary systems, arXiv:1408.2202v1 [math.PR] (2014).
- Ditlev Monrad and Walter Philipp, Nearby variables with nearby conditional laws and a strong approximation theorem for Hilbert space valued martingales, Probab. Theory Related Fields 88 (1991), no. 3, 381–404. MR 1100898, DOI 10.1007/BF01418867
- F. Móricz, Moment inequalities and the strong laws of large numbers, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 35 (1976), no. 4, 299–314. MR 407950, DOI 10.1007/BF00532956
- Bernard Petit, Le théorème limite central pour des sommes de Riesz-Raĭkov, Probab. Theory Related Fields 93 (1992), no. 4, 407–438 (French, with English summary). MR 1183885, DOI 10.1007/BF01192715
- Walter Philipp, 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. MR 1249469, DOI 10.1090/S0002-9947-1994-1249469-5
- Volker Strassen, Almost sure behavior of sums of independent random variables and martingales, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 315–343. MR 0214118
- Georgi E. Shilov, Linear algebra, Revised English edition, Dover Publications, Inc., New York, 1977. Translated from the Russian and edited by Richard A. Silverman. MR 0466162
- Shigeru Takahashi, On the distribution of values of the type $\sum f(q^{k}t)$, Tǒhoku Math. J. (2) 14 (1962), 233–243. MR 0145274, DOI 10.2748/tmj/1178244114
- S. C. Zaremba, Some applications of multidimensional integration by parts, Ann. Polon. Math. 21 (1968), 85–96. MR 235731, DOI 10.4064/ap-21-1-85-96
Additional Information
- Katusi Fukuyama
- Affiliation: Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
- MR Author ID: 256708
- Email: fukuyama@math.kobe-u.ac.jp
- Received by editor(s): November 27, 2017
- Received by editor(s) in revised form: May 2, 2018
- Published electronically: February 1, 2019
- Additional Notes: This research was partially supported by JSPS KAKENHI 16K05204 and 15KT0106. It was also partially supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located at Kyoto University.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 1193-1211
- MSC (2010): Primary 42A55, 60F05
- DOI: https://doi.org/10.1090/tran/7772
- MathSciNet review: 3968800
Dedicated: Dedicated to Professor Norio Kôno on his $80$th birthday