# Simultaneous Pointwise Majorization for Mixed Tail Processes with Applications in Gaussian Chaos and Ergodic Diffusions

arXiv: https://arxiv.org/abs/2609.01576 · HTML: https://mlsanity.com/paper/2609.01576/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 45 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Builds on (most cited first, 10 of 10)

- [Towards Optimal Problem Dependent Generalization Error Bounds in Statistical Learning Theory](https://mlsanity.com/paper/2011.06186/) (2020)
- [Contextual Online Decision Making with Infinite-Dimensional Functional Regression](https://mlsanity.com/paper/2501.18359/) (2025)
- [Wild refitting for black box prediction](https://mlsanity.com/paper/2506.21460/) (2025)
- [Majorizing Measures for the Optimizer](https://mlsanity.com/paper/2012.13306/) (2020)
- [Non-Asymptotic Pointwise and Worst-Case Bounds for Classical Spectrum Estimators](https://mlsanity.com/paper/2303.11908/) (2023)
- [Interleaved Resampling and Refitting: Data and Compute-Efficient Evaluation of Black-Box Predictors](https://mlsanity.com/paper/2603.14218/) (2026)
- [Perturbing the Derivative: Doubly Wild Refitting for Model-Free Evaluation of Opaque Machine Learning Predictors](https://mlsanity.com/paper/2511.18789/) (2025)
- [Pointwise Complexity for Gaussian Fields: Upper Envelopes, Algorithmic Lower Bounds, and Separation](https://mlsanity.com/paper/2606.07931/) (2026)
- [Upper Confidence Bounds for the Prediction Error of Kernel Ridge Regression via Gaussian Refitting](https://mlsanity.com/paper/2607.28846/) (2026)
- [Majorizing Measures, Codes, and Information](https://mlsanity.com/paper/2305.02960/) (2023)

## Abstract (arXiv)

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,ldots,dm, and orders α1,ldots,αm>0, beginalign* mathbbP|Zt-Zs|>sumj=1m u1/αjdj(s,t)≤ 2e-u,s,t∈ T. endalign* For ambient priors μ1,ldots,μm, let vj(t):=dj(t,t0), Phij(t):=int04vj(t)(logfrac1μj(Bdj(t,r)))1/αjdr. We prove that, forall δ∈(0,1), with probability at least 1-δ, simultaneously for all t∈ T, beginalign* |Zt|≤ Cm,boldsymbolαsumj=1mPhij(t)+vj(t)(log(e/δ))1/αj. endalign* Here boldsymbolα:=(α1,ldots,αm) and Cm,boldsymbolα 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 citetxu2026 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.