Limit behavior of symmetric random walks with a membrane

Let $\{X(k), k\in \mathbb {Z}_+\}$ be a random walk in $\mathbb {Z}$. Assume that its transition probabilities coincide with those of a symmetric random walk with unit steps throughout except for a fixed neighborhood of zero. The weak convergence of the sequence of normalized walks $\{X_n(k) = n^{-1/2} X(nk),k\geqslant 0\}_{n\geqslant 1}$ is proved. The main result generalizes a Harrison and Shepp theorem on the weak convergence to a skew Brownian motion in the case where the symmetricity of the random walk fails at a single point. All possible limits for the corresponding random walks are described.