  The paper is concerned with a new approach for the recurrence property of the
oscillating process on $\mathbb{Z}$ in Kemperman's sense. In the case when the
random walk is ascending on $\mathbb{Z}^-$ and descending on $\mathbb{Z}^+$, we
determine the invariant measure of the embedded process of successive crossing
times and then prove a necessary and sufficient condition for recurrence.
Finally, we make use of this result to show that the general oscillating
process is recurrent under some H{\"o}lder-typed moment assumptions.
