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

Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition