A lower bound in the law of the iterated logarithm for general lacunary series
This was first proved for Bernoulli random variables by Khintchine [K] and grew out of the efforts of several authors to determine the exact rate of convergence in Borel’s theorem on normal numbers. Although the terms in a lacunary trigonometric series are not independent random variables, it is evidenced by many results in analysis which give central limit theorem type behavior or LILs for lacunary trigonometric series, that they exhibit many of the same properties. For example, Salem and Zygmund [SZ] consider the situation when the Xk in Kolmogorov’s theorem are replaced by the functions ak cos(nkθ) on [−π, π], where the ak are real and the nk are integers satisfying the lacunarity condition: there exists a number q so that