A Note on the Finite-Exceedance Dependence Principle for Sum--Maximum Asymptotic Independence

Abstract

Asymptotic independence between sum-type and maximum-type statistics arises repeatedly across many areas of probability and statistics, particularly in high-dimensional testing and random matrix theory. Motivated by the shared structure underlying existing proofs, which is obscured by their model-specific formulations, this note isolates a common intermediate principle. We introduce a finite-exceedance dependence principle: if the finite-exceedance dependence error vanishes at every fixed order and high-order exceedances are mildly controlled, then a finite-order Bonferroni argument yields asymptotic independence between the sum and the maximum. We further give an independent-proxy criterion for verifying this: for each finite set of exceedances, it suffices to construct an exactly independent proxy for the sum-type statistic whose approximation error is negligible in aggregate. Local deletion under dependency graphs and Gaussian regression and residualization are two examples. The framework gives modular derivations of existing sum--max independence results.

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BibTeX

@misc{shi2026note,
  title = {{A Note on the Finite-Exceedance Dependence Principle for Sum--Maximum Asymptotic Independence}},
  author = {Hongjian Shi},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03351},
  eprint = {2611.03351},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03351},
}

RIS

TY  - UNPB
TI  - A Note on the Finite-Exceedance Dependence Principle for Sum--Maximum Asymptotic Independence
AU  - Hongjian Shi
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03351
UR  - https://arxiv.org/abs/2611.03351
AB  - Asymptotic independence between sum-type and maximum-type statistics arises repeatedly across many areas of probability and statistics, particularly in high-dimensional testing and random matrix theory. Motivated by the shared structure underlying existing proofs, which is obscured by their model-specific formulations, this note isolates a common intermediate principle. We introduce a finite-exceedance dependence principle: if the finite-exceedance dependence error vanishes at every fixed order and high-order exceedances are mildly controlled, then a finite-order Bonferroni argument yields asymptotic independence between the sum and the maximum. We further give an independent-proxy criterion for verifying this: for each finite set of exceedances, it suffices to construct an exactly independent proxy for the sum-type statistic whose approximation error is negligible in aggregate. Local deletion under dependency graphs and Gaussian regression and residualization are two examples. The framework gives modular derivations of existing sum--max independence results.
ER  -

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