The central limit theorem for Riesz–Raikov sums II

Fukuyama, Katusi

doi:10.1090/tran/7772

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