Mary Weiss

The Law of the Iterated Logarithm for Lacunary Trigonometric Series
Transactions of the American Mathematical Society, Vol. 91, No. 3 (June 1959), 444-469

Introduction

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

(1.1)       

be a lacunary trigonometric series, that is to say, one such that n_k+1/n_k > q > 1 for all k. We write

If, for N —>+∞,

(1.2)           and    

then we have, for almost all x,

(1.3)       

 

In this theorem the a_k and b_k are real and the n_k 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 n_k 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 Σe^in_kx and that the n_k 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 achnowledgments 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.

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