AI 中文总结
该研究针对可分希尔伯特空间中的鞅，证明了时间一致的自归一化贝内特不等式，扩展至补偿标记点过程，得到重对数律等推论，优化了有限维下的相关界。
AI 中文摘要
我们针对可分希尔伯特空间中满足$M_0=0$且范数增量不超过1的鞅$(M_n)_{n\text{≥}0}$，证明了时间一致的自归一化贝内特不等式。记$V_n$为其可预测协方差过程，$h(u)=(1+u)\log(1+u)-u$为贝内特速率函数，我们证明：对任意正则化参数$\rho>0$，过程$\exp\left\{\rho\\,h\\!\left(\frac{\left\lVert (V_n+\rho I)^{-1/2}M_n\right\rVert}{\sqrt{\rho}}\right)-\frac12\log\det\left(I+\rho^{-1}V_n\right)\right\},\quad n\geq0$是初值为1的非负上鞅，其中$\det$为弗雷德霍姆行列式。利用维勒不等式可得到$\lVert (V_n+\rho I)^{-1/2}M_n\rVert$的时间一致贝内特和伯恩斯坦界。该结果允许条件协方差增量为无限秩；在有限维情形下，所得边界优于现有的鞅变换及基于行列式的变分界。相同构造可扩展至带有界跳的补偿标记点过程，通过对$\rho$混合这些上鞅可实现对正则化参数的同时控制。其推论包括可分希尔伯特空间中正则化椭球半径的重对数律上界，以及有限维情形下谱敏感的有限时间界和未正则化半径$\lVert V_n^{-1/2}M_n\rVert$的重对数律上界，该上界的常数1在对应类中是最优的。我们还得到了依赖于$\operatorname{tr}(V_n)$和$\lVert V_n\rVert_{\mathrm{op}}$的鞅范数$\lVert M_n\rVert$的时间一致伯恩斯坦不等式。
英文摘要
We prove time-uniform self-normalised Bennett inequalities for a martingale $(M_n)_{n\geq0}$ in a separable Hilbert space, with $M_0=0$ and increments bounded in norm by one. Writing $V_n$ for its predictable covariance process and $h(u)=(1+u)\log(1+u)-u$ for the Bennett rate function, we show that, for every regularisation parameter $ρ>0$, the process \[\exp\left\{ρ\,h\!\left(\frac{\left\lVert (V_n+ρI)^{-1/2}M_n\right\rVert}{\sqrtρ}\right)-\frac12\log\det\left(I+ρ^{-1}V_n\right)\right\},\qquad n\geq0,\] is a nonnegative supermartingale with initial value one, where $\det$ is the Fredholm determinant. Ville's inequality yields time-uniform Bennett and Bernstein bounds for $\lVert (V_n+ρI)^{-1/2}M_n\rVert$. The result permits conditional covariance increments of infinite rank; in finite dimensions, the resulting boundaries sharpen existing martingale-transform and determinant-based variational bounds. The same construction extends to compensated marked point processes with bounded jumps. Mixing these supermartingales over $ρ$ gives simultaneous control over the regularisation parameter. Consequences include an upper law of the iterated logarithm for the regularised ellipsoidal radius in separable Hilbert spaces and, in finite dimensions, spectrum-sensitive finite-time bounds and an upper law of the iterated logarithm for the unregularised radius $\lVert V_n^{-1/2}M_n\rVert$, whose constant $1$ is sharp over the class. We also obtain a time-uniform Bernstein inequality for the martingale norm $\lVert M_n\rVert$ with dependence on $\operatorname{tr}(V_n)$ and $\lVert V_n\rVert_{\mathrm{op}}$.
