Hypothesis Testing For Dependent Variables With Unbalanced Data

This research develops a unified theoretical framework to evaluate the asymptotic convergence of multiple dependent random variables with unbalanced data toward a common population mean (μ) under weak dependence conditions (α-mixing). Three approaches based on the law of large numbers are employed: triangular arrays (TAC) and weighted correlation sums (WSC) for the strong version, and mixingale processes (MPC) for the weak version. This aims to verify whether deviations from μ are statistically insignificant in the limit. A theorem of strict metric equivalence is proved under specific necessary and sufficient conditions, including an important corollary on exponential dependence. The proposed framework extends the Neyman-Pearson lemma for dependent variables and establishes uniform confidence bounds, incorporating careful consideration of the measurable structure and associated filtrations. Rigorous Type I error control with dependence is developed, and Bayesian extensions are established to incorporate prior information about the dependence structure. The fundamental relationships between convergence, errors, and Bayesian probabilities are made explicit through non-asymptotic inequalities. The work demonstrates that the three approaches are asymptotically equivalent under α-mixing dependence with polynomial decay rate, providing a unified theoretical basis for convergence analysis in weak dependence scenarios. The proposed architecture has been designed with sufficient flexibility to incorporate additional methodologies in future developments.