Central Limit Theorems for Weighted Sums of Dependent Random Vectors in Hilbert Spaces via the Theory of the Regular Variation

In this paper, based on the theory of regularly varying functions we study central limit theorems for the weighted sum $$S_n=\sum _{j=1}^{m_n}c_{nj}X_{nj}$$ , where $$(X_{nj};1\le j \le m_n,n\ge 1)$$ is a Hilbert-space-valued identically distributed martingale difference array and $$(c_{nj};1\le j \le m_n,n\ge 1)$$ is an array of real numbers. As an application, we present a central limit theorem for moving average processes of martingale differences.