Weak laws of large numbers for maximal weighted sums of random variables

Abstract

In this paper, we establish a weak law of large numbers for a class of weighted sums of random variables introduced by Jajte (2003). The obtained method allows us to deduce a generalized version of the Marcinkiewicz-Zygmund weak law of large numbers and to strengthen several known results, such as those of Gut (2004) and Naderi et al. (2018). Finally, an application to randomly indexed sums is presented.