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The multivariate functional de Jong CLT

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Abstract

We prove a multivariate functional version of de Jong’s CLT (J Multivar Anal 34(2):275–289, 1990) yielding that, given a sequence of vectors of Hoeffding-degenerate U-statistics, the corresponding empirical processes on [0, 1] weakly converge in the Skorohod space as soon as their fourth cumulants in \(t=1\) vanish asymptotically and a certain strengthening of the Lindeberg-type condition is verified. As an application, we lift to the functional level the ‘universality of Wiener chaos’ phenomenon first observed in Nourdin et al. (Ann Probab 38(5):1947–1985, 2010).

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Acknowledgements

This work is part of the first author’s habilitation thesis at Heinrich Heine Universität Düsseldorf. The research leading to this paper was supported by the FNR grant FoRGES (R-AGR-3376-10) at Luxembourg University. This work is also part of project Stein-ML that has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101024264. We thank an anonymous referee for several useful remarks. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Döbler, C., Kasprzak, M. & Peccati, G. The multivariate functional de Jong CLT. Probab. Theory Relat. Fields 184, 367–399 (2022). https://doi.org/10.1007/s00440-022-01114-3

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