Pointwise Majorization for sub-Weibull and Mixed Tail Processes with Applications in Quadratic Chaos and Ergodic Diffusions

Source abstract

Classical chaining controls an indexed stochastic process through a single worst-case bound, which can obscure substantial variation across the index set. We establish the first simultaneous pointwise majorization theory for Banach-valued processes with sub-Weibull or two-metric mixed-tail increments. For an anchored sub-Weibull process on a separable index space, write $v(t):=d(t,t_0)$. Given a reference measure $μ$, the envelope at $t$ is governed by the pointwise Fernique-Talagrand functional of order $α$, $Φ_{μ,d}^{(α)}(t):=\int_0^{4v(t)}(\log\frac{1}{μ(B_d(t,r))})^{1/α}dr$. $\forall δ\in(0,1)$, we obtain that $$\mathbb{P}(\|Z_t\|\lesssim\{Φ_{μ,d}^{(α)}(t)+v(t)(\log(e/δ))^{1/α}\},\forall t)\ge 1-δ.$$ Our bound is determined by the pointwise complexity $Φ_{μ,d}^{(α)}$ rather than a global quantity. The result holds for every $α>0$ and does not involve dyadic logarithmic terms from peeling. For mixed tail processes, with fixed measures $μ_1,μ_2$ and $v_j(t):=d_j(t,t_0)$, $Φ_j(t):=\int_0^{4v_j(t)}(\log\frac{1}{μ_j(B_{d_j}(t,r))})^{1/α_j}dr, j=1,2$, for any $δ\in(0,1)$, we show that $$\mathbb{P}(\|Z_t\|\lesssim\sum_{j=1}^2\{Φ_j(t)+v_j(t)(\log\frac{e}δ)^{1/α_j}\},\forall t)\ge 1-δ.$$ Although the two regimes are coupled in the mixed tail condition, each retains its own pseudo-metric, reference measure, pointwise Fernique-Talagrand functional, and tail exponent. The proof tracks the index-wise costs of measure-generated admissible chains and synchronizes them through a nested common refinement. For applications, we derive matrix-specific bounds for centered quadratic chaos under pseudo-metrics induced by the operator and Frobenius norms, and observable-specific finite-time bounds for diffusion empirical processes.