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

Ta Cong Son () and Le Van Dung ()
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Ta Cong Son: VNU University of Science, Vietnam National University
Le Van Dung: The University of Da Nang - University of Science and Education

Journal of Theoretical Probability, 2022, vol. 35, issue 2, 988-1012

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

Keywords: Central limit theorems; Martingale differences; Hilbert space; Weighted sum; Regular variation; 60F05; 60G46; 62J05 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01079-4

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