Harmut Holtzwart (hartmut.holzwart@allianz.de) asks

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Related question not treated in Schmulands article:

What is known about the distribution of

Z = e_1 + 1/3 e_2 + 1/5 e_3 + 1/7 e_4 + ....

with e_i  i.i.d. +/-1?
>>

If X = sum from 1 to oo of e_k (1/k) is the random variable for the sum of the random harmonic series, then clearly Z = X_1 - 2*X_2 (where X_1 and X_2 are independent and have the same distribution as X).