Kolmogorov-Type Maximal Inequalities for Independent and Dependent Negative Binomial Random Variables: Sharp Bounds, Sub-Exponential Refinements, and Applications to Overdispersed Count Data

TL;DR

This work develops Kolmogorov-type maximal inequalities for sums of Negative Binomial variables under both independence and Poisson-Gamma dependent structures. The authors derive a sharp Markov-type deviation bound for independent NB variables and a Kolmogorov-type bound with explicit Tweedie NB2 control limits, linking dispersion to GLM surveillance through $\mathrm{Var}(X_i)=\mu_i+\kappa_i\mu_i^2$. For dependent counts from a shared Gamma mixing variable, they introduce a basic Kolmogorov bound and a novel sub-exponential Bernstein-type refinement that exploits the hierarchical Poisson-Gamma structure to achieve exponential tail decay. Moment-matched Monte Carlo experiments show a 55% reduction in mean maximum deviation under dependence, with an analytical explanation grounded in the shared latent factor. An epidemiological application using NB2 parameters calibrated from COVID-19 data demonstrates the practical utility of these maximal-inequality results for monitoring overdispersed counts in public health.

Abstract

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.

Figures (8)

• Figure 1: Mean-variance relationship $\mathrm{Var}(X)=\mu+\mu^2/r$ for NB variables with $p \in \{0.3, 0.5, 0.7\}$. Quadratic growth demonstrates the Tweedie NB2 structure.
• Figure 2: Overdispersion index $\mathrm{Var}(X)/\mathbb{E}[X]=1+(1-p)/(rp)$ versus $r$. The $1/r$ decay confirms convergence to Poisson behavior as $r\to\infty$.
• Figure 3: Kolmogorov control limits for $n=20$ independent heterogeneous NB variables over 100 simulations. Control limit $\lambda_\alpha=\sqrt{V_n/\alpha}$ (red) with $\alpha=0.05$ achieves empirical exceedance rate of 4%, validating Lemma \ref{['lem:kolmogorov_indep']}.
• Figure 4: Theoretical bounds for representative parameters: Kolmogorov (independent), Kolmogorov (dependent), and Bernstein (dependent). The Bernstein bound exhibits exponential decay, while Kolmogorov bounds decay as $\lambda^{-2}$. Dashed line: 5% control limit.
• Figure 5: Distribution of $\max_{1\le k\le n}|S_k|$ for moment-matched cases (n=20, 2,000 replications). Despite identical marginal moments, the dependent case shows substantially smaller deviations.

Theorems & Definitions (14)

• Lemma 2.1: Markov-Type Deviation Inequality for NB Sample Mean
• proof
• Remark 2.2: Connection with the Tweedie--NB2 model
• Lemma 3.1: Kolmogorov-Type Bound for Heterogeneous NB Variables
• proof
• Corollary 3.2: Tweedie--NB Control Limit
• Theorem 4.1: Kolmogorov-Type Inequality with Shared Gamma Mixing
• proof
• Theorem 4.2: Sub-Exponential Bernstein Bound for Dependent NB Partial Sums
• proof