Strong convergence theorems for coordinatewise negatively associated random vectors in Hilbert space

Area of Science:

• Probability theory
• Functional analysis
• Stochastic processes

Background:

• Coordinatewise negatively associated (CNA) random vectors are a significant class of dependent random variables.
• Hilbert spaces provide a fundamental framework for studying infinite-dimensional vector spaces and their properties.
• Convergence theorems are crucial for understanding the asymptotic behavior of random processes.

Purpose of the Study:

• To establish novel strong convergence theorems for weighted sums of CNA random vectors in Hilbert spaces.
• To extend and improve upon existing convergence results in the literature.
• To correct and enhance previous findings concerning random vector convergence.

Main Methods:

• Utilizing techniques from probability theory and functional analysis.
• Developing new inequalities and lemmas specific to CNA random vectors in Hilbert spaces.
• Applying established methods for proving strong convergence in stochastic analysis.

Main Results:

• Established strong convergence theorems for weighted sums of CNA random vectors.
• Demonstrated improvement and extension of results by Huan et al. (2014).
• Corrected and enhanced the convergence results presented by Ko (2017).

Conclusions:

• The established theorems provide a more comprehensive understanding of the convergence properties of weighted sums of CNA random vectors.
• This work advances the theoretical framework for dependent random variables in Hilbert spaces.
• The findings offer a corrected and improved basis for future research in this area.