  In this paper we establish a multivariate exchangeable pairs approach within
the framework of Stein's method to assess distributional distances to
potentially singular multivariate normal distributions. By extending the
statistics into a higher-dimensional space, we also propose an embedding method
which allows for a normal approximation even when the corresponding statistics
of interest do not lend themselves easily to Stein's exchangeable pairs
approach. To illustrate the method, we provide the examples of runs on the line
as well as double-indexed permutation statistics.
