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 ≤ 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