## 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.