A functional central limit theorem for particle systems under polynomial growth conditions, with applications to statistics
Abstract
We establish a functional central limit theorem (CLT) for fluctuations of time-integrated functionals of empirical measures arising from interacting McKean--Vlasov particle systems. Our framework allows for coefficients with unbounded spatial growth and dependence on arbitrarily high-order moments of the measure variable, extending the scope of the existing fluctuation theory. The proof uses a novel approach combining linear functional differentiation with new well-posedness results and estimates for perturbed backward parabolic equations in polynomially weighted Schauder spaces, yielding a martingale-type representation of the fluctuations with an asymptotically negligible remainder. The limiting process is shown to be centred Gaussian with an explicit covariance structure. As a statistical application, we derive the asymptotic theory underlying hypothesis tests for both the temporal constancy of population characteristics and parametric restrictions on the drift coefficient.
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BibTeX
@misc{podolskij2026functional,
title = {{A functional central limit theorem for particle systems under polynomial growth conditions, with applications to statistics}},
author = {Mark Podolskij and Fabrice Wunderlich},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34347},
eprint = {2610.34347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34347},
}
RIS
TY - UNPB
TI - A functional central limit theorem for particle systems under polynomial growth conditions, with applications to statistics
AU - Mark Podolskij
AU - Fabrice Wunderlich
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34347
UR - https://arxiv.org/abs/2610.34347
AB - We establish a functional central limit theorem (CLT) for fluctuations of time-integrated functionals of empirical measures arising from interacting McKean--Vlasov particle systems. Our framework allows for coefficients with unbounded spatial growth and dependence on arbitrarily high-order moments of the measure variable, extending the scope of the existing fluctuation theory. The proof uses a novel approach combining linear functional differentiation with new well-posedness results and estimates for perturbed backward parabolic equations in polynomially weighted Schauder spaces, yielding a martingale-type representation of the fluctuations with an asymptotically negligible remainder. The limiting process is shown to be centred Gaussian with an explicit covariance structure. As a statistical application, we derive the asymptotic theory underlying hypothesis tests for both the temporal constancy of population characteristics and parametric restrictions on the drift coefficient.
ER -