Limit behavior of the “horizontal-vertical” random walk and some extensions of the Donsker–Prokhorov invariance principle

We consider a two-dimensional random walk that moves in the horizontal direction on the half-plane $\{y>x\}$ and in the vertical direction on the half-plane $\{y\le x\}$. The limit behavior (as the time interval between two steps and the size of each step tend to zero) of this “horizontal-vertical” random walk is investigated. In order to solve this problem, we prove an extension of the Donsker–Prokhorov invariance principle. The extension states that the discrete-time stochastic integrals with respect to the appropriately renormalized one-dimensional random walk converge in distribution to the corresponding stochastic integral with respect to a Brownian motion. This extension enables us to construct a discrete-time approximation of the local time of a Brownian motion. We also provide discrete-time approximations of skew Brownian motions.