Strong limit theorems for weighted partial sums of widely orthant dependent series and probability inequalities with applications to first order autoregressive models

This study investigates widely orthant dependent random variables, establishing probability inequalities and deriving their key properties, which are essential for the development of strong limit theorems. These random variables define a specific structure of dependence that plays a crucial role in understanding various statistical challenges in both theoretical and applied contexts. Building on these foundational results, we extend the Khinchin-Kolmogorov convergence theorem and the three series convergence theorem to the case of widely orthant dependent random variables. Additionally, we derive strong limit theorems for weighted sums of widely orthant dependent random variables, extending existing findings in the literature . The core theoretical contributions are demonstrated through their application to first order autoregressive models, emphasizing the relevance of these results in time series analysis. Numerical simulations are conducted to verify and visualize the theoretical findings, showcasing their applicability and robustness. These results provide insights into the behavior of widely orthant dependent random variables and contribute to the advancement of statistical theory.