Open Access
2021 Stein’s method of exchangeable pairs in multivariate functional approximations
Christian Döbler, Mikołaj J. Kasprzak
Author Affiliations +
Electron. J. Probab. 26: 1-50 (2021). DOI: 10.1214/21-EJP587


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

Funding Statement

Mikołaj Kasprzak was supported by the FNR grant FoRGES (R-AGR-3376-10) at Luxembourg University.


The authors would like to thank Gesine Reinert, Giovanni Peccati and Alison Etheridge for helpful discussions and comments on the early versions of this work.


Download Citation

Christian Döbler. Mikołaj J. Kasprzak. "Stein’s method of exchangeable pairs in multivariate functional approximations." Electron. J. Probab. 26 1 - 50, 2021.


Received: 1 June 2020; Accepted: 30 January 2021; Published: 2021
First available in Project Euclid: 23 March 2021

Digital Object Identifier: 10.1214/21-EJP587

Primary: 60B10 , 60F17
Secondary: 0B12 , 60E05 , 60E15 , 60J65

Keywords: Exchangeable pairs , Functional convergence , multivariate processes , Stein’s method , U-statistics

Vol.26 • 2021
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