Asymptotic Theory for Combining Dependent $p$-Values for Global Hypothesis Testing

Abstract

Combining $p$-values is a fundamental procedure in global hypothesis testing. In modern high-dimensional settings, however, component $p$-values often exhibit complex dependence and rely on asymptotic approximations rather than exact finite-sample uniform distributions. This paper establishes a unified asymptotic theory for weighted transformation statistics that decouples marginal finite-sample approximation error from the joint dependence structure. We also provide sufficient conditions based on conditional probability bounds to verify the joint-tail conditions. Utilizing this framework, we derive explicit dimension-growth and correlation rates for test statistics operating under asymptotic Gaussian and chi-square calibrations. For non-exact finite-sample statistics, we analyze standardized weighted sums, demonstrating how Cram\'er moderate deviations control relative tail error. Analytical examples demonstrate why both marginal and joint conditions are mathematically indispensable for valid global inference under dependence, and numerical experiments confirm that our asymptotic framework maintains accurate finite-sample size control at extreme significance levels.