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Area of Science:

  • Probability Theory
  • Stochastic Processes
  • Mathematical Statistics

Background:

  • Weak laws of large numbers are fundamental in probability theory, describing the convergence of sample averages.
  • Pairwise NQD (negatively dependent) random variables exhibit a specific type of dependence structure with implications for convergence theorems.
  • Investigating random variables with infinite means presents unique challenges in establishing convergence properties.

Purpose of the Study:

  • To extend the theory of weak laws of large numbers to weighted pairwise NQD random variables.
  • To analyze the behavior of these variables under conditions of infinite mean.
  • To establish precise bounds for normalized weighted sums of these random variables.

Main Methods:

  • Application of techniques from probability theory to analyze sums of dependent random variables.
  • Development of methods to handle infinite mean cases within the framework of weak laws.
  • Derivation of almost sure convergence results using inequalities and limit theorems.

Main Results:

  • Establishment of weak laws of large numbers for weighted pairwise NQD random variables with infinite means.
  • Determination of almost sure upper and lower bounds for specific normalized weighted sums.
  • Demonstration of convergence properties under generalized dependence conditions.

Conclusions:

  • The findings extend existing limit theorems for dependent random variables.
  • The results provide a deeper understanding of the probabilistic behavior of weighted pairwise NQD variables.
  • The established bounds are valuable for statistical inference and modeling involving such variables.