Some mean convergence theorems for arrays of rowwise pairwise negative quadrant dependent random variables

Area of Science:

• Probability Theory
• Stochastic Processes
• Mathematical Statistics

Background:

• Existing research on convergence of random variables, including Chandra's work.
• The importance of dependence structures in random variable arrays.
• Limitations of previous studies regarding pairwise negative quadrant dependence.

Purpose of the Study:

• To establish conditions for the convergence in mean to 0 of weighted averages for rowwise pairwise negative quadrant dependent random variables.
• To establish conditions for the convergence in mean to 0 of normed and centered row sums for such arrays.
• To extend existing convergence theorems in probability theory.

Main Methods:

• Analysis of arrays of random variables with rowwise pairwise negative quadrant dependence.
• Development of theoretical conditions for convergence in mean.
• Construction of counterexamples to illustrate limitations.

Main Results:

• Conditions provided for weighted averages to converge in mean to 0, extending Chandra's results.
• Conditions provided for normed and centered row sums to converge in mean to 0.
• Demonstration that results hold even for independent random variables within rows.

Conclusions:

• The established conditions provide new insights into the convergence properties of dependent random variables.
• The findings are significant even when random variables in each row are independent.
• Counterexamples highlight the necessity of the pairwise negative quadrant dependence assumption and show that almost sure convergence is not guaranteed.