Simultaneous Pointwise Majorization for Mixed Tail Processes with Applications in Gaussian Chaos and Ergodic Diffusions
Abstract
Classical chaining controls an indexed stochastic process through a single worst-case bound and can therefore obscure substantial variation across the index set. We develop the first simultaneous pointwise majorization theory for Banach-valued processes with finite-metric mixed-tail increments. Suppose that an anchored process (Zt)t∈ T satisfies, for some integer m≥1, pseudo-metrics d1,…,dm, and orders α1,…,αm>0, ℙ{|Zt−Zs|>∑j=1m u1/αjdj(s,t)}≤ 2e−u,s,t∈ T. For ambient priors μ1,…,μm, let vj(t):=dj(t,t0), Φj(t):=∫04vj(t)(log1/μj(Bdj(t,r)))1/αjdr. We prove that, ∀ δ∈(0,1), with probability at least 1−δ, simultaneously for all t∈ T, |Zt|≤ Cm,α∑j=1m{Φj(t)+vj(t)(log(e/δ))1/αj}. Here α:=(α1,…,αm) and Cm,α depend only on m and these tail orders. The result subsumes single-metric sub-Weibull processes of every positive order as the case m=1. In the Gaussian setting, it sharpens the pointwise upper bound of \citet{xu2026} by eliminating the logarithmic terms generated by dyadic peeling. The proof retains the index-wise costs of measure-generated admissible chains and synchronizes the regimes through a nested common refinement. Finally, we apply our theorems to stationary diffusion empirical processes and decoupled Gaussian chaos to obtain simultaneous pointwise envelope bounds, which can further be applied to other statistics problems.
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BibTeX
@misc{hu2026simultaneous,
title = {{Simultaneous Pointwise Majorization for Mixed Tail Processes with Applications in Gaussian Chaos and Ergodic Diffusions}},
author = {Haichen Hu and David Simchi-Levi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01576},
eprint = {2609.01576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01576},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Simultaneous Pointwise Majorization for Mixed Tail Processes with Applications in Gaussian Chaos and Ergodic Diffusions
AU - Haichen Hu
AU - David Simchi-Levi
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.01576
UR - https://arxiv.org/abs/2609.01576
AB - Classical chaining controls an indexed stochastic process through a single worst-case bound and can therefore obscure substantial variation across the index set. We develop the first simultaneous pointwise majorization theory for Banach-valued processes with finite-metric mixed-tail increments. Suppose that an anchored process (Z_t)_t∈ T satisfies, for some integer m≥1, pseudo-metrics d_1,…,d_m, and orders α_1,…,α_m>0, ℙ{|Z_t−Z_s|>∑_j=1^m u^1/α_jd_j(s,t)}≤ 2e^−u,s,t∈ T. For ambient priors μ_1,…,μ_m, let v_j(t):=d_j(t,t_0), Φ_j(t):=∫_0^4v_j(t)(log1/μ_j(B_d_j(t,r)))^1/α_jdr. We prove that, ∀ δ∈(0,1), with probability at least 1−δ, simultaneously for all t∈ T, |Z_t|≤ C_m,α∑_j=1^m{Φ_j(t)+v_j(t)(log(e/δ))^1/α_j}. Here α:=(α_1,…,α_m) and C_m,α depend only on m and these tail orders. The result subsumes single-metric sub-Weibull processes of every positive order as the case m=1. In the Gaussian setting, it sharpens the pointwise upper bound of \citet{xu2026} by eliminating the logarithmic terms generated by dyadic peeling. The proof retains the index-wise costs of measure-generated admissible chains and synchronizes the regimes through a nested common refinement. Finally, we apply our theorems to stationary diffusion empirical processes and decoupled Gaussian chaos to obtain simultaneous pointwise envelope bounds, which can further be applied to other statistics problems.
ER -