Recurrence of Fourier sums

Ulrich Haböck
(University of Vienna)

Abstract: We consider random sums $S_n = \sum_{k=1}^n X_k e^{i\alpha k}$ where we assume the angle $\alpha$ to be fixed and the (real valued) coefficients $X_k$ originating from a stationary process. Such sums can be regarded as (stationary) random walk in the semi-direct product $S^1\ltimes R^2$.
The first part of the talk will be about a recurrence criterion of such random walks which roughly speaking states that transient (i.e. the opposite of recurrent) random walks must escape to infinity at a certain rate. As a consequence it turns out that if our coefficient process $(X_k)_{k\geq 1}$ is asymptotical independent (and has finite second moments) then for almost every angles $\alpha$ (with respect to the Lebesgue measure) the corresponding random walk $S_n$ is recurrent, no matter how slow the $X_k$ become independent from one another.