On lacunary series with random gaps (Q485026)

It is a well-known fact that lacunary trigonometric sums exhibit many properties which are typical for sums of independent, identically distributed (i.i.d.) random variables. In the sub-lacunary case, this almost independence-property generally breaks down, and the probabilisitic behavior of the corresponding trigonometric sums is much more complicated. In the paper under review, the author considers dilated sums of the form \[ \sum_k f(S_k x), \tag{1} \] where \(f\) is a 1-periodic function, and \(S_k=\sum_{j=1}^k X_j\) is a sum of i.i.d. random variables. Under the assumption that \(f\) is Lipschitz-continuous and that the \(X_j\)'s have a bounded density, the author obtains a Strassen-type functional law of the iterated logarithm for the sum (1), which holds almost surely on the probability space on which the \(X_j\)'s are defined. In other words, while for slowly growing (deterministic) sequences \((n_k)\) the system \(\sum_k f(n_k x)\) usually fails to show random behavior, a randomized argument shows that such slowly growing sequences for which \(\sum_k f(n_k x)\) acts randomly do exist, and that such sequences are actually ``typical'' (in an appropriate randomized model). This observation is in accordance with earlier work on this subject, but the author's results are stronger and more precise than earlier ones (although they require, as noted, the summands \(X_j\) which generate \((S_k)\) to posses a density, which means that in this model the resulting sequence \((S_k)\) is not a sequence of integers). The proof uses methods of \textit{P. Schatte} (see, e.g., [Math. Nachr. 115, 275--281 (1984; Zbl 0557.60008)] and [Probab. Theory Relat. Fields 77, No. 2, 167--178 (1988; Zbl 0619.60032)]).