Weak Laws of Large Numbers for sequences of random variables with infinite rth moments

Abstract

Let $${(X_n;n \geq 1)}$$ be a sequence of independent random variables with infinite rth absolute moments for some $${0 < r < 2}$$ . We investigate weak laws of large numbers for the weighted sum $${S_n = \sum_{j=1}^{m_n}c_{nj}X_j}$$ , where $${(c_{nj};1 \leq j \leq m_n,n \geq 1)}$$ is an array of real numbers. As illustrative examples, we obtain a weak law of large numbers of extended Pareto–Zipf distributions and generalized Feller Game. Furthermore, these results are applied to study weak law of large numbers of moving average sums of a sequence of i.i.d. random variables.