Stein’s method of exchangeable pairs in multivariate functional approximations

In this paper we develop a framework for multivariate functional approximation by suitable Gaussian process via an exchangeable pairs coupling that satisfies approximate linear regression property, thereby building on work Barbour (1990) and Kasprzak (2020). We demonstrate the applicability of our results applying them to joint subgraph counts in Erd?s-Renyi random graph model one hand vectors weighted, degenerate U-processes other hand. As concrete instance latter class examples, provide bound vector success runs different lengths which, even situation just single run, would be outside scope existing theory.