Kolmogorov-Type Maximal Inequalities for Independent and Dependent Negative Binomial Random Variables: Sharp Bounds, Sub-Exponential Refinements, and Applications to Overdispersed Count Data
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914410871652352 |
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| author | Doumas, Aristides V. Spektor, S. |
| author_facet | Doumas, Aristides V. Spektor, S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18184 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kolmogorov-Type Maximal Inequalities for Independent and Dependent Negative Binomial Random Variables: Sharp Bounds, Sub-Exponential Refinements, and Applications to Overdispersed Count Data Doumas, Aristides V. Spektor, S. Statistics Theory Probability Methodology 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. |
| title | Kolmogorov-Type Maximal Inequalities for Independent and Dependent Negative Binomial Random Variables: Sharp Bounds, Sub-Exponential Refinements, and Applications to Overdispersed Count Data |
| topic | Statistics Theory Probability Methodology |
| url | https://arxiv.org/abs/2602.18184 |