Complete convergence of randomly weighted END sequences and its application

Area of Science:

• Probability Theory
• Stochastic Processes
• Control Systems Engineering

Background:

• Extended negatively dependent (END) random variables are a key structure in probability theory.
• Understanding convergence properties of partial sums is crucial for statistical analysis.
• State observers for linear-time-invariant systems require robust convergence guarantees.

Purpose of the Study:

• To investigate the complete convergence of partial sums for randomly weighted END random variables.
• To establish results on complete moment convergence and the strong law of large numbers for END structures.
• To apply these findings to analyze the convergence of state observers in dynamic systems.

Main Methods:

• Utilizing techniques from probability theory to analyze partial sums of dependent random variables.
• Developing novel inequalities and lemmas tailored for extended negative dependence.
• Applying established convergence theorems to the specific context of END random variables.

Main Results:

• Established theorems on the complete convergence of partial sums for randomly weighted END random variables.
• Obtained new results for complete moment convergence and the strong law of large numbers under END conditions.
• Demonstrated the applicability of these convergence properties to state observer analysis.

Conclusions:

• The study provides significant theoretical advancements in the convergence properties of dependent random variables.
• The obtained results offer a stronger theoretical foundation for analyzing state observer convergence.
• This work extends and generalizes existing findings in the field of dependent random variables and their applications.