a decomposition approach to the asymptotic distribution of the sample variance in discrete models

Understanding the asymptotic behavior of the sample variance is important in statistical theory and inference. while classical results provide chi-square limiting distributions for continuous populations, discrete random variables often exhibit non-classical behavior, complicating both theoretical analysis and practical applications. in this paper, we propose a decomposition of the sample variance for discrete random variables into two theoretically tractable components, enabling a finer characterization of their stochastic structure. we derive the asymptotic distributions of these components under a specific condition, revealing that certain components converge to chi-square laws - a phenomenon previously observed only in bernoulli and binomial models with success probability one-half. extensive monte carlo simulations confirm the accuracy of the theoretical approximations and demonstrate their relevance even for moderate sample sizes. these results provide new insights into the interplay between discreteness, variance decomposition, and asymptotic behavior, extending classical chi-square asymptotics to a broader class of discrete models.