  The paper establishes a functional version of the Hoeffding combinatorial
central limit theorem. First, a pre-limiting Gaussian process approximation is
defined, and is shown to be at a distance of the order of the Lyapounov ratio
from the original random process. Distance is measured by comparison of
expectations of smooth functionals of the processes, and the argument is by way
of Stein's method. The pre-limiting process is then shown, under weak
conditions, to converge to a Gaussian limit process. The theorem is used to
describe the shape of random permutation tableaux.
