Lp maximal inequalities for Gaussian subordination with applications to Breuer--Major--Donsker principles
Abstract
In this paper, we establish an L2 maximal inequality for partial sums of Gaussian-subordinated random variables. More precisely, let f:ℝ→ℝ belong to L2(γ) and have Hermite rank at least d, where γ denotes the standard Gaussian measure. If the covariance matrix of the underlying Gaussian family, with standard normal marginals, satisfies a uniform dth-power row bound K, then the L2 norm of the maximal partial sum is bounded by Cd,K√n |f|L2(γ). The constant Cd,K is independent of (f,n) and depends on the covariance structure only through K. As a consequence, our L2 maximal inequality controls Hermite tails uniformly in the path norm. In discrete time, this yields the Breuer--Major--Donsker principle under the sole finite-variance assumption, removing both the additional L2+-integrability assumption of Nourdin and Nualart (Probab. Theory Related Fields, 2020) and the prediction-theoretic or decimation assumptions of Mansanarez, Poly, and Zheng (arXiv:2607.11469). In continuous time, we likewise obtain the Breuer--Major--Donsker principle under finite variance alone, removing the additional moment assumption of Campese, Nourdin, and Nualart (Ann. Probab., 2020). For the nonstationary self-similar setting considered there, we remove the same extra moment assumption in the subcritical regime, while a separate leading-chaos argument yields the corresponding logarithmically normalized critical limit. [Abstract shorten to meet arXiv requirement]
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BibTeX
@misc{milesis2026maximal,
title = {{$L^p$ maximal inequalities for Gaussian subordination with applications to Breuer--Major--Donsker principles}},
author = {Dionysis Milesis and Guangqu Zheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09404},
eprint = {2610.09404},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09404},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - L^p maximal inequalities for Gaussian subordination with applications to Breuer--Major--Donsker principles
AU - Dionysis Milesis
AU - Guangqu Zheng
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09404
UR - https://arxiv.org/abs/2610.09404
AB - In this paper, we establish an L^2 maximal inequality for partial sums of Gaussian-subordinated random variables. More precisely, let f:ℝ→ℝ belong to L^2(γ) and have Hermite rank at least d, where γ denotes the standard Gaussian measure. If the covariance matrix of the underlying Gaussian family, with standard normal marginals, satisfies a uniform dth-power row bound K, then the L^2 norm of the maximal partial sum is bounded by C_d,K√n |f|_L^2(γ). The constant C_d,K is independent of (f,n) and depends on the covariance structure only through K. As a consequence, our L^2 maximal inequality controls Hermite tails uniformly in the path norm. In discrete time, this yields the Breuer--Major--Donsker principle under the sole finite-variance assumption, removing both the additional L^2+-integrability assumption of Nourdin and Nualart (Probab. Theory Related Fields, 2020) and the prediction-theoretic or decimation assumptions of Mansanarez, Poly, and Zheng (arXiv:2607.11469). In continuous time, we likewise obtain the Breuer--Major--Donsker principle under finite variance alone, removing the additional moment assumption of Campese, Nourdin, and Nualart (Ann. Probab., 2020). For the nonstationary self-similar setting considered there, we remove the same extra moment assumption in the subcritical regime, while a separate leading-chaos argument yields the corresponding logarithmically normalized critical limit. [Abstract shorten to meet arXiv requirement]
ER -