This paper develops Kolmogorov-type maximal inequalities for sums of Negative Binomial random variables under both independence and dependence structures. For independent heterogeneous Negative Binomial variables we derive sharp Markov-type deviation inequalities and Kolmogorov-type bounds expressed in terms of Tweedie dispersion parameters, providing explicit control limits for NB2 generalized linear model monitoring. For dependent count data arising through a shared Gamma mixing variable, we establish a \emph{sub-exponential Bernstein-type refinement} that exploits the Poisson-Gamma hierarchical structure to yield exponentially decaying tail probabilities -- this refinement is new in the literature. Through moment-matched Monte Carlo experiments ($n=20$, 2{,}000 replications), we document a 55\% reduction in mean maximum deviation under appropriate dependence structures, a stabilization effect we explain analytically. A concrete epidemiological application with NB2 parameters calibrated from COVID-19 surveillance data demonstrates practical utility. These results materially advance the applicability of classical maximal inequalities to overdispersed and dependent count data prevalent in public health, insurance, and ecological modeling.

翻译：本文在独立与相依两种相依结构下，建立了负二项随机变量和的Kolmogorov型极大不等式。对于独立异质负二项变量，推导了以Tweedie离散参数表示的Markov型精确偏差不等式和Kolmogorov型界，为NB2广义线性模型监测提供了显式控制限。对于由共享Gamma混合变量产生的相依计数数据，建立了基于Poisson-Gamma层次结构的次指数Bernstein型精化不等式，该精化可产生指数衰减的尾概率——这一精化在文献中尚属首次。通过矩匹配蒙特卡洛实验（$n=20$，2,000次重复），我们发现在适当相依结构下平均最大偏差降低55%，并从解析角度解释了这一稳定效应。一项基于COVID-19监测数据校准的NB2参数的具体流行病学应用展示了其实用价值。这些结果实质性地推进了经典极大不等式在公共卫生、保险和生态建模中普遍存在的过离散与相依计数数据中的适用性。