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

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.