The Central Limit Theorem for Subsequences in Probabilistic Number Theory

Aistleitner, Christoph ; Elsholtz, Christian

Canadian journal of mathematics, Tome 64 (2012) no. 6, pp. 1201-1221

Résumé

Let ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ be an increasing sequence of positive integers, and let $f\left( x \right)$ be a real function satisfying 1 $$f\left( x+1 \right)=f\left( x \right),\int\limits_{0}^{1}{f\left( x \right)}dx=0,\text{Va}{{\text{r}}_{\left[ 0,1 \right]}}f<\infty $$ If ${{\lim }_{k\to \infty }}\frac{{{n}_{k+1}}}{{{n}_{k}}}=\infty $ the distribution of 2 $$\frac{\sum\nolimits_{k=1}^{N}{f\left( {{n}_{k}}x \right)}}{\sqrt{N}}$$ converges to a Gaussian distribution. In the case $$1<\underset{k\to \infty }{\mathop{\lim \inf }}\,\,\frac{{{n}_{k+1}}}{{{n}_{k}}},\,\,\underset{k\to \infty }{\mathop{\text{lim}\,\text{sup}}}\,\,\frac{{{n}_{k+1}}}{{{n}_{k}}}<\infty$$ there is a complex interplay between the analytic properties of the function $f$ , the number-theoretic properties of ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ , and the limit distribution of (2).In this paper we prove that any sequence ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ satisfying $\lim {{\sup }_{k\to \infty }}{{n}_{k+1}}/{{n}_{k}}=1$ contains a nontrivial subsequence ${{\left( {{m}_{k}} \right)}_{k\ge 1}}$ such that for any function satisfying (1) the distribution of $$\frac{\sum\nolimits_{k=1}^{N}{f\left( {{m}_{k}}x \right)}}{\sqrt{N}}$$ converges to a Gaussian distribution. This result is best possible: for any $\varepsilon >0$ there exists a sequence ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ satisfying lim $\underset{k\to \infty }{\mathop{\sup }}\,\frac{{{n}_{k+1}}}{{{n}_{k}}}\le 1+\varepsilon$ such that for every nontrivial subsequence ${{\left( {{m}_{k}} \right)}_{k\ge 1}}$ of ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$ .Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

Détail

DOI : 10.4153/CJM-2011-074-1

Mots-clés : 60F05, 42A55, 11D04, 05C55, 11K06, central limit theorem, lacunary sequences, linear Diophantine equations, Ramsey type theorem

Bibliographie

[1] [1] Aistleitner, C., Diophantine equations and the LIL for the discrepancy of sublacunary sequences. Illinois J. Math. 53(2009), 785–815. Google Scholar

[2] [2] Aistleitner, C., On the law of the iterated logarithm for the discrepancy of Lacunary sequences. Trans. Amer. Math. Soc. 362(2010), 5967–5982. Google Scholar | DOI

[3] [3] Aistleitner, C. and Berkes, I., On the central limit theorem for f (nkx). Probab˙Theory Related Fields 146(2010), 267–289. Google Scholar | DOI

[4] [4] Berkes, I., A central limit theorem for trigonometric series with small gaps. Z.Wahrsch. Verw. Gebiete 47(1979), 157–161. Google Scholar | DOI

[5] [5] Berkes, I., Philipp, W. and Tichy, R. F., Empirical processes in probabilistic number theory: the LIL for the discrepancy of (nk!) mod 1. Illinois J. Math. 50(2006), 107–145 (electronic). Google Scholar

[6] [6] Berkes, I., Metric discrepancy results for sequences ﹛nkx﹜ and Diophantine equations. In: Diophantine approximation, Dev. Math. 16, Springer Wien New York, Vienna, 2008, 95–105. Google Scholar

[7] [7] Bobkov, S. G. and Götze, F., On the central limit theorem along subsequences of noncorrelated observations. Teor. Veroyatnost. i Primenen. 48(2003), 745–765. Google Scholar

[8] [8] Bobkov, S. G. and Götze, F., Concentration inequalities and limit theorems for randomized sums. Probab. Theory Related Fields 137(2007), 49–81. Google Scholar | DOI

[9] [9] Burkill, H. and Mirsky, L., Monotonicity. J. Math. Anal. Appl. 41(1973), 391–410. Google Scholar | DOI

[10] [10] Chatterji, S. D., A principle of subsequences in probability theory: the central limit theorem. Advances in Math. 13(1974), 31–54; correction, ibid. 14(1974), 266–269. Google Scholar | DOI

[11] [11] Drmota, M. and Tichy, R. F., Sequences, discrepancies and applications. Lecture Notes in Math. 1651, Springer–Verlag, Berlin, 1997. Google Scholar

[12] [12] Dupain, Y. and Sós, V. T., On the one-sided boundedness of discrepancy-function of the sequence ﹛n﹜. Acta Arith. 37(1980), 363–374. Google Scholar

[13] [13] Erdös, P. and Gál, I. S., On the law of the iterated logarithm. I, II. Nederl. Akad.Wetensch. Proc. Ser. A 58 = Indag. Math. 17(1955), 65–76, 77–84. Google Scholar

[14] [14] Fukuyama, K., A central limit theorem for trigonometric series with bounded gaps. Probab. Theory Related Fields 149(2011), 139–148. Google Scholar | DOI

[15] [15] Fukuyama, K., A central limit theorem and a metric discrepancy result for sequences with bounded gaps. In: Dependence in Probability, Analysis and Number Theory, Kendrick Press, 2010, 233–246. Google Scholar

[16] [16] Fukuyama, K. and Nakata, K., A metric discrepancy result for the Hardy–Littlewood–Pólya sequences. Monatsh. Math. 160(2010), 41–49. Google Scholar | DOI

[17] [17] Fukuyama, K. and Petit, B., Le théorème limite central pour les suites de R. C. Baker. Ergodic Theory Dynam. Systems 21(2001), 479–492. Google Scholar

[18] [18] Gapoškin, V. F., Lacunary series and independent functions. Uspehi Mat. Nauk 21(1966), 3–82. Google Scholar

[19] [19] Gapoškin, V. F., The central limit theorem for certain weakly dependent sequences. Teor. Verojatnost. i Primenen. 15(1970), 666–684. Google Scholar

[20] [20] Kac, M., Probability methods in some problems of analysis and number theory. Bull. Amer. Math. Soc. 55(1949), 641–665. Google Scholar | DOI

[21] [21] Kesten, H., The discrepancy of random sequences ﹛kx﹜. Acta Arith. 10(1964/1965), 183–213. Google Scholar

[22] [22] Kuipers, L. and Niederreiter, H., Uniform distribution of sequences. Pure and Applied Mathematics, Wiley-Interscience [John Wiley &amp; Sons], New York, 1974. Google Scholar

[23] [23] Něsetřil, J., Ramsey theory. In: Handbook of combinatorics, Vol. 1, 2, Elsevier, Amsterdam, 1995, 1331–1403. Google Scholar

[24] [24] Philipp, W., Limit theorems for lacunary series and uniform distribution mod1. Acta Arith. 26(1974/75), 241–251. Google Scholar

[25] [25] Philipp, W., 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), 705–727. Google Scholar | DOI

[26] [26] Salem, R. and Zygmund, A., On lacunary trigonometric series. Proc. Nat. Acad. Sci. U. S. A. 33(1947), 333–338. Google Scholar | DOI

[27] [27] Salem, R. and Zygmund, A., La loi du logarithme itéré pour les séries trigonométriques lacunaires. Bull. Sci. Math. (2) 74(1950), 209–224. Google Scholar

[28] [28] Sanders, T., On Roth's theorem on progressions. Ann. of Math. 174(2011), 619–636. Google Scholar | DOI

[29] [29] Schoissengeier, J., On the discrepancy of (n). Acta Arith. 44(1984), 241–279. Google Scholar

[30] [30] Schoissengeier, J., A metrical result on the discrepancy of (n). Glasgow Math. J. 40(1998), 393–425. Google Scholar | DOI

[31] [31] Szemerédi, E., On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27(1975), 199–245. Google Scholar

[32] [32] Takahashi, S., A gap sequence with gaps bigger than the Hadamards. Tôhoku Math. J. 13(1961), 105–111. Google Scholar | DOI

[33] [33] Zygmund, A., Trigonometric series. Vol. I, II. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1979 edition. Google Scholar