Laws of large numbers for a sequence of pairwise independent stochastically dominated random variables

Abstract

Classical limit theorems in probability theory are typically based on the assumption that random variables are independent and identically distributed. Fundamental laws of large numbers, such as the weak and strong laws of large numbers by Kolmogorov and the weak law by Marcinkiewicz-Zygmund, were established under these assumptions. However, in practice, random variables often exhibit more complex relationships that require more flexible assumptions to accurately capture the nature of real-world phenomena. In this paper, instead of assuming complete independence and identical distribution, the author establishes results based on the assumptions of pairwise independence and stochastic domination of the sequence of random variables. This allows for an extension of classical laws of large numbers, making them more suitable for complex real-world situations. In addition, the author establishes laws of large numbers for weighted sums of random variables, further generalizing the classical results of Kolmogorov and Marcinkiewicz-Zygmund.