The purpose of this paper is to prove the following theorem of "The Law of the Iterated Logarithm" for lacunary trigonometric series.
Theorem 1. Let
be a lacunary trigonometric series, that is to say, one such that nk+1/nk > q > 1 for all k. We write
If, for N —>+∞,
then we have, for almost all x,
In this theorem the ak and bk are real and the nk are positive but not necessarily integral. The latter point is important for applications.
The Law of the Iterated Logarithm for lacunary series has already been treated in the literature. In the first place, Salem and Zygmund have shown that under the hypotheses (1.2) we have (1.3) with "<" instead of "=." They also presuppose that the nk are integers. A complete proof of (1.3) was given later by Erdös and Gál under the restriction however that S is of the form Σeinkx and that the nk are integers. In the proof which follows we use some ideas from these two papers and also from the classical paper of Kolmogoroff on the Law of the Iterated Logarithm for independent random variables. More detailed acknowledgments will be given at the proper places.
I am grateful to Professor Zygmund for calling my attention to the problem and for helping me with suggestions.