  We show that almost any one-dimensional projection of a suitably scaled
random walk on a hypercube, inscribed in a hypersphere, converges weakly to an
Ornstein-Uhlenbeck process as the dimension of the sphere tends to infinity. We
also observe that the same result holds when the random walk is replaced with
spherical Brownian motion. This latter result can be viewed as a "functional"
generalisation of Poincar\'e's observation for projections of uniform measure
on high dimensional spheres; the former result is an analogous generalisation
of the Bernoulli-Laplace central limit theorem. Given the relation of these two
classic results to the central limit theorem for convex bodies, the modest
results provided here would appear to motivate a functional generalisation.
