发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对混合尾过程开发了首个逐点同时控制理论，改进了高斯场景下的逐点上界，并将其应用于平稳扩散经验过程与解耦高斯混沌以获得相关包络界。
AI 中文摘要
经典的链式方法通过单一最坏情况界控制索引随机过程，因此会掩盖索引集内的大量变异。我们开发了首个针对具有有限度量混合尾增量的巴拿赫空间值过程的逐点同时控制理论。假设锚定过程 $(Z_t)_{t\in T}$ 满足：对某个整数 $m\ge1$、伪度量 $d_1,\ldots,d_m$ 及阶数 $\alpha_1,\ldots,\alpha_m>0$，有 $\mathbb{P}\{\\|Z_t-Z_s\\|>\sum_{j=1}^m u^{1/\alpha_j}d_j(s,t)\}\le 2e^{-u}$（其中 $s,t\in T$）。对于环境先验 $\mu_1,\ldots,\mu_m$，令 $v_j(t):=d_j(t,t_0)$，$\Phi_j(t):=\int_0^{4v_j(t)}(\log\frac{1}{\mu_j(B_{d_j}(t,r))})^{1/\alpha_j}dr$。我们证明，对任意 $\delta\in(0,1)$，以至少 $1-\delta$ 的概率，对所有 $t\in T$ 同时成立 $\\|Z_t\\|\le C_{m,\boldsymbol\alpha}\sum_{j=1}^m\{\Phi_j(t)+v_j(t)(\log(e/\delta))^{1/\alpha_j}\}$，其中 $\boldsymbol\alpha:=(\alpha_1,\ldots,\alpha_m)$，$C_{m,\boldsymbol\alpha}$ 仅依赖于 $m$ 和这些尾阶。该结果包含了所有正阶的单度量次威布尔过程（对应 $m=1$ 的情况）；在高斯场景下，它消除了文献[xu2026]中由二进剥离产生的对数项，从而改进了逐点上界。证明保留了测度生成的容许链的索引相关代价，并通过嵌套公共细化同步了各区域。最后，我们将定理应用于平稳扩散经验过程和解耦高斯混沌，以获得逐点同时包络界，该界可进一步应用于其他统计问题。
英文摘要
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 $(Z_t)_{t\in T}$ satisfies, for some integer $m\ge1$, pseudo-metrics $d_1,\ldots,d_m$, and orders $α_1,\ldots,α_m>0$, \begin{align*} \mathbb{P}\{\|Z_t-Z_s\|>\sum_{j=1}^m u^{1/α_j}d_j(s,t)\}\le 2e^{-u},s,t\in T. \end{align*} For ambient priors $μ_1,\ldots,μ_m$, let $v_j(t):=d_j(t,t_0), Φ_j(t):=\int_0^{4v_j(t)}(\log\frac{1}{μ_j(B_{d_j}(t,r))})^{1/α_j}dr$. We prove that, $\forall δ\in(0,1)$, with probability at least $1-δ$, simultaneously for all $t\in T$, \begin{align*} \|Z_t\|\le C_{m,\boldsymbolα}\sum_{j=1}^m\{Φ_j(t)+v_j(t)(\log(e/δ))^{1/α_j}\}. \end{align*} Here $\boldsymbolα:=(α_1,\ldots,α_m)$ and $C_{m,\boldsymbolα}$ 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 \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.
