the wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. this is known as donsker's theorem. like the random walk, the wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. unlike the random walk, it is scale invariant, meaning that