  A classical limit theorem of stochastic process theory concerns the sample
cumulative distribution function (CDF) from independent random variables. If
the variables are uniformly distributed then these centered CDFs converge in a
suitable sense to the sample paths of a Brownian Bridge. The so-called
Hungarian construction of Komlos, Major and Tusnady provides a strong form of
this result. In this construction the CDFs and the Brownian Bridge sample paths
are coupled through an appropriate representation of each on the same
measurable space, and the convergence is uniform at a suitable rate.
  Within the last decade several asymptotic statistical-equivalence theorems
for nonparametric problems have been proven, beginning with Brown and Low
(1996) and Nussbaum (1996). The approach here to statistical-equivalence is
firmly rooted within the asymptotic statistical theory created by L. Le Cam but
in some respects goes beyond earlier results.
  This talk demonstrates the analogy between these results and those from the
coupling method for proving stochastic process limit theorems. These two
classes of theorems possess a strong inter-relationship, and technical methods
from each domain can profitably be employed in the other. Results in a recent
paper by Carter, Low, Zhang and myself will be described from this perspective.
