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In this paper we develop a framework for multivariate functionalapproximation by a suitable Gaussian process via an exchangeable pairs couplingthat satisfies a suitable approximate linear regression property, therebybuilding on work by Barbour (1990) and Kasprzak (2020). We demonstrate theapplicability of our results by applying it to joint subgraph counts in anErd\H{o}s-Renyi random graph model on the one hand and to vectors of weighted,degenerate $U$-processes on the other hand. As a concrete instance of thelatter class of examples, we provide a bound for the functional approximationof a vector of success runs of different lengths by a suitable Gaussian processwhich, even in the situation of just a single run, would be outside the scopeof the existing theory.

Stein's method of exchangeable pairs in multivariate functional approximations