Convergence of weighted sums of pairwise independent stochastically dominated random variables

Tóm tắt

Limit theorems play an important role in probability theory and have numerous applications in statistics. Among them, the weak and strong laws of large numbers established by Kolmogorov and Marcinkiewicz–Zygmund for sequences of independent and identically distributed random variables are classical results of fundamental significance. These results have attracted considerable attention and have been extended in various directions under more general settings. In this paper, we further develop these classical results for sequences of pairwise independent random variables. The assumption of identical distribution is replaced by a weaker condition, in which the random variables are stochastically dominated by another random variable with a heavy-tailed distribution. Based on this setting, we establish limit theorems for weighted sums of random variables, thereby extending the applicability of the laws of large numbers to more general conditions. By applying these results, we investigate the convergence for a nonparametric regression estimator.