A metric discrepancy result for lacunary sequences
Authors:
Katusi Fukuyama and Tetsujin Watada
Journal:
Proc. Amer. Math. Soc. 140 (2012), 749754
MSC (2010):
Primary 11K38; Secondary 60F15
Published electronically:
June 23, 2011
MathSciNet review:
2869060
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We prove that every value greater than or equal to can be a constant appearing in the law of the iterated logarithm for discrepancies of a lacunary sequence satisfying the Hadamard gap condition.
 1.
Christoph
Aistleitner, On the law of the iterated logarithm
for the discrepancy of lacunary sequences, Trans. Amer. Math. Soc. 362 (2010), no. 11, 5967–5982. MR 2661504
(2012a:11109), 10.1090/S000299472010050263
 2.
C. Aistleitner, I. Berkes, R. Tichy, On permutation of HardyLittlewoodPolya sequences, Trans. Amer. Math. Soc. (to appear).
 3.
István
Berkes and Walter
Philipp, The size of trigonometric and Walsh series and uniform
distribution 𝑚𝑜𝑑\1, J. London Math. Soc. (2)
50 (1994), no. 3, 454–464. MR 1299450
(96e:11099), 10.1112/jlms/50.3.454
 4.
K.
Fukuyama, An asymptotic property of gap series, Acta Math.
Hungar. 97 (2002), no. 3, 257–264. MR 1933732
(2003i:42013), 10.1023/A:1020859112807
 5.
K.
Fukuyama, The law of the iterated logarithm for discrepancies of
{𝜃ⁿ𝑥}, Acta Math. Hungar. 118
(2008), no. 12, 155–170. MR 2378547
(2008m:60049), 10.1007/s1047400762018
 6.
Katusi
Fukuyama and Keisuke
Nakata, A metric discrepancy result for the
HardyLittlewoodPólya sequences, Monatsh. Math.
160 (2010), no. 1, 41–49. MR 2610311
(2011h:11080), 10.1007/s0060500800515
 7.
K. Fukuyama, A central limit theorem and a metric discrepancy result for sequence with bounded gaps, Dependence in Probability, Analysis and Number Theory, A volume in memory of Walter Philipp, Kendrick Press, Heber City, UT (2010), 233246.
 8.
K. Fukuyama, N. Hiroshima, Metric discrepancy results for subsequences of , Monatsh. Math. (to appear).
 9.
Walter
Philipp, Limit theorems for lacunary series and uniform
distribution 𝑚𝑜𝑑\1, Acta Arith.
26 (1974/75), no. 3, 241–251. MR 0379420
(52 #325)
 10.
Walter
Philipp, A functional law of the iterated logarithm for empirical
distribution functions of weakly dependent random variables, Ann.
Probability 5 (1977), no. 3, 319–350. MR 0443024
(56 #1397)
 1.
 C. Aistleitner, On the law of the iterated logarithm for the discrepancy of lacunary sequences, Trans. Amer. Math. Soc., 362 (2010) 59675982. MR 2661504
 2.
 C. Aistleitner, I. Berkes, R. Tichy, On permutation of HardyLittlewoodPolya sequences, Trans. Amer. Math. Soc. (to appear).
 3.
 I. Berkes, W. Philipp, The size of trigonometric and Walsh series and uniform distribution mod 1, Jour. London Math. Soc. (2), 50 (1994) 454464. MR 1299450 (96e:11099)
 4.
 K. Fukuyama, An asymptotic property of gap series, Acta Math. Hungar., 97 (2002) 257264. MR 1933732 (2003i:42013)
 5.
 K. Fukuyama, The law of the iterated logarithm for discrepancies of , Acta Math. Hungar., 118 (2008) 155170. MR 2378547 (2008m:60049)
 6.
 K. Fukuyama, K. Nakata, A metric discrepancy result for the HardyLittlewoodPólya sequences, Monatsh. Math., 160 (2010) 4149. MR 2610311
 7.
 K. Fukuyama, A central limit theorem and a metric discrepancy result for sequence with bounded gaps, Dependence in Probability, Analysis and Number Theory, A volume in memory of Walter Philipp, Kendrick Press, Heber City, UT (2010), 233246.
 8.
 K. Fukuyama, N. Hiroshima, Metric discrepancy results for subsequences of , Monatsh. Math. (to appear).
 9.
 W. Philipp, Limit theorems for lacunary series and uniform distribution mod 1, Acta Arith., 26 (1975) 241251. MR 0379420 (52:325)
 10.
 W. Philipp, A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables, Ann. Probab., 5 (1977) 319350. MR 0443024 (56:1397)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
11K38,
60F15
Retrieve articles in all journals
with MSC (2010):
11K38,
60F15
Additional Information
Katusi Fukuyama
Affiliation:
Department of Mathematics, Kobe University, Rokko, Kobe, 6578501 Japan
Email:
fukuyama@math.kobeu.ac.jp
Tetsujin Watada
Affiliation:
Department of Mathematics, Kobe University, Rokko, Kobe, 6578501 Japan
DOI:
http://dx.doi.org/10.1090/S000299392011109407
Keywords:
Discrepancy,
lacunary sequence,
law of the iterated logarithm
Received by editor(s):
November 26, 2010
Received by editor(s) in revised form:
December 8, 2010
Published electronically:
June 23, 2011
Additional Notes:
The first author was supported in part by KAKENHI 19204008.
Communicated by:
Richard C. Bradley
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
