On the law of the iterated logarithm for trigonometric series with bounded gaps

• 成果类型: Article
• 署名作者: Aistleitner, Christoph; Fukuyama, Katusi
• 署名单位: Graz University of Technology; Kobe University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-011-0378-z
• 发表日期: 2012
• 页码: 607-620
• 关键词: lacunary series; LIMIT-THEOREMS; discrepancy

摘要

Let (n(k)) k >= 1 be an increasing sequence of positive integers. Bobkov and Gotze proved that if the distribution of cos 2 pi n(1)x + ... + cos 2 pi nNx/root N converges to a Gaussian distribution, then the value of the variance is bounded from above by 1/2 - lim sup k/(2n(k)). In particular it is impossible that for a sequence (n(k)) k >= 1 with bounded gaps (i.e. n(k+1) - n(k) <= c for some constant c) the distribution of (1) converges to a Gaussian distribution with variance sigma(2) = 1/2 or larger. In this paper we show that the situation is considerably different in the case of the law of the iterated logarithm. We prove the existence of an increasing sequence of positive integers satisfying n(k+1) - n(k) <= 2 lim sup(N ->infinity) Sigma(N)(k=1) cos 2 pi n(k)x/root 2Nlog log N = +infinity a.e.