Functional law of the iterated logarithm for lacunary trigonometric and some gap series (Q2367107)

The series indicated in the title are treated as random series on the interval [0,1] endowed with a probability measure satisfying some regularity conditions (mainly with the usual Lebesgue measure). The author proves the functional (or Strassen's) law of the iterated logarithm for (i) weakly multiplicative systems, (ii) lacunary trigonometric series, (iii) gap series, i.e., series of the form \(\sum^ \infty_{k=1}a_ kf(\beta_ kx+\gamma_ k)\), where \(\beta_{k+1}/\beta_ k\to\infty\) as \(k\to\infty\), \(\{a_ k\}\) and \(f\) satisfy some natural conditions, and \(\{\gamma_ k\}\) is arbitrary. Among others, the author proves more general theorems than those proved earlier by \textit{I. Berkes} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 34, 319-345 (1976; Zbl 0403.60004) and ibid., 347-365 (1976; Zbl 0403.60005)] and \textit{S. Takahashi} [Tôhoku Math. J., II. Ser. 31, 439-451 (1979; Zbl 0407.60022)].