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

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.