Projected random walk on sphere converges to Brownian motion?

Question

Random walks on a unit sphere can be generated by the walks in $\mathbb R^3$ projected onto the sphere. I read in a paper that it can be rigorously proven that the random walks converge to Brownian motion on the sphere. The hints given is that the proof can be done by following the steps of the proof of Donsker’s theorem. That means, one needs to check that the corresponding finite-dimensional distributions of these random walks converge as $\Delta t \rightarrow 0$, and then prove tightness of the family of probability measures induced by these random walks. Since I am a beginner stochastic student, can someone prove it rigorously or explain the steps in more detail?