A functional limit theorem for lattice oscillating random walk

The paper is devoted to an invariance principle for Kemperman’s model of oscillating random walk on $\mathbb{Z}$. This result appears as an extension of the invariance principal theorem for classical random walks on $\mathbb{Z}$ or reflected random walks on ${\mathbb{N}}_0$. Relying on some natural Markov sub-process which takes into account the oscillation of the random walks between ${\mathbb{Z}}^{-}$ and ${\mathbb{Z}}^{+}$, we first construct an aperiodic sequence of renewal operators acting on a suitable Banach space and then apply a powerful theorem proved by S. Gouëzel.