The complete moment convergence for CNA random vectors in Hilbert spaces

Area of Science:

• Probability Theory
• Stochastic Analysis
• Functional Analysis

Background:

• Coordinatewise negatively associated (CNA) random vectors are a significant class of dependent random variables.
• Complete moment convergence is a crucial concept in probability theory, providing strong convergence rates.
• Existing research has established complete moment convergence in various settings, but extensions to Hilbert spaces for CNA sequences were needed.

Purpose of the Study:

• To establish the complete moment convergence for sequences of coordinatewise negatively associated random vectors in Hilbert spaces.
• To extend the existing results on complete moment convergence to a more general setting (Hilbert spaces).
• To generalize the Baum-Katz type theorem to the context of complete moment convergence for these specific random vectors.

Main Methods:

• Utilizing techniques from probability theory and stochastic analysis.
• Applying methods relevant to random vectors in infinite-dimensional spaces (Hilbert spaces).
• Building upon and extending established theorems related to moment convergence and dependent random variables.

Main Results:

• The paper successfully establishes the complete moment convergence for sequences of coordinatewise negatively associated random vectors in Hilbert spaces.
• The established result extends the findings of Ko (2016) concerning complete moment convergence to Hilbert spaces.
• The study generalizes the Baum-Katz type theorem (Huan et al., 2014) to the domain of complete moment convergence for CNA random vectors.

Conclusions:

• The findings provide a significant advancement in the understanding of convergence properties for dependent random vectors in Hilbert spaces.
• This work bridges existing theoretical gaps and offers a more comprehensive framework for analyzing complete moment convergence.
• The generalization of the Baum-Katz theorem contributes to the broader theory of probability in Banach and Hilbert spaces.