This may be unhelpful, but have you tried Feller's books, which after all are later. For example, Volume 2 of "An introduction to probability theory and its applications" has Chapter 12 (Random walks in R1) with a section on "Distribution of ladder epochs", which I think deals with sign changes.

Thanks a lot. It seems, I have misunderstood the question, posed in Feller's 1945 paper. I'm interested in refinements of the Law of Iterated Logarithm. It states that the well-known expression must infinitely oscillate between -1 and +1. Could something general be said about the frequency and wave length of these oscillations? 

It may be that Feller's question can't  be answered, but maybe someone has worked further on it since Feller's death in about 1970. 

But the work on "ladder epochs" should be somewhat useful, since it relates to returns to zero and the gap between returns to zero can reasonably be interpreted as half-a-wavelength. Thus the results should allow evaluation of an expected value of this half-wavelength or to know if the expected value exists. Of course, some extra argument might be needed to convert theory for "return to zero" to that for "change of sign". 

However, this doesn't immediately relate to a Fourier-type analysis of the process.

Have you been able to find the results by Erdös that Feller refers to as being unpublished, as this offers a way forward? As you can tell, this is not an field in which I have any relevant experience.